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University of Bath
PHD
Design and analysis of AC machines for traction purposes.
Coles, Philip Charles
Award date:1984
Awarding institution:University of Bath
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DESIGN AND ANALYSIS OF AC MACHINES
FOR TRACTION PURPOSES
subm itted by
Philip Charles Coles B .S c .
for the degree of P h .D . of the University of Bath
1 9 8 4
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7 M A Y
ABSTRACT
Interest Is continually being shown in the rep lacem e n t of variab le speed
DC m ach in es with an equivalent inverter fed AC m ach ine. This is as true
In the field of highly rated m ach ines for traction use, as in the field of
sm aller industria l drives.
In the follow ing w ork, a g en era l design m ethod is presented that is
suitable for the design of induction , and slip ring synchronous m achines
of the round rotor or sa lien t pole type. The m ethod is based upon
m ach ine m odels that em ploy surface quantities. This type of m odel c learly
displays the in teraction betw een flux and cu rren t, and is readily adaptable
for design use, as the am ount of detail requ ired is kept to a m inim um .
S everal designs a re p resented for induction and synchronous m achines
that satisfy the requ irem ents of a traction m otor for use in a high speed
locom otive. P erfo rm an ce p red ictions, based on sinusoidal supply
considera tions and operating under two com m only used control schem es
are shown. The traction m otors satisfy the m ain overall requ irem ent for a
m inim um size and w eight design .
In p ractice the AC traction m otors would be supplied by a variable
frequency inverter. In view of th is, an analysis of the perfo rm ance of the
most su itab le induction and synchronous m otor designs is p resented ,
when each is being supplied with an inverter of the preferred type. Two
inverters a re c o n s id e red , one of the constant voltage type, and one of the
constant c u rren t type. C om puter m odels are used to predict the m achine
vo ltag e , c u rre n t and torque waveform s when both Inverters a re operated
In the 120 d e g re e conduction m ode.
E xperim enta l results a re shown, to verify the com puter m odel of the
c u rren t so u rce in verter. A 5KVA laboratory squirrel cag e induction
m ach in e is used for this purpose, in conjunction with a fo rce m easuring
platform that en ab les the steady state torque pulsations to be recorded .
ACKNOW LEDGEMENTS
The au thor w ishes to s incere ly thank his supervisor. Dr. M .J . B aich in .
for his help and e n co u rag em en t during the m any discussions that w ere
had throughout the course of the work described in this thesis.
The au thor a lso w ishes to thank Professor J. F. Eastham for m aking
ava ilab ie the w ide ran g e of school fac ilities , and M rs. A. Baichin for her
help in p re p arin g the thesis.
F ina lly , the fin a n c ia l support of the S c ien ce and E ng ineering R esearch
C ouncil and G EC Tractio n is g ratefu lly acknow ledged.
INDEX
Abstract 2
Acknow ledgem ents 4
Index 5
List of Symbols 7
C hapter 1 G enera l Introduction 13
C hapter 2 The design of induction and synchronous m ach in es 22based on a surface equivalent m odel
2. 1 introduction 23
2 .2 M achine m odels and equivalent c ircu its 26
2 . 2 . 1 The induction m ach ine 262 . 2 . 2 The synchronous m ach in e 30
2 .3 A consideration of the g eo m etric and m ag n etic 35circu it aspects of the design m ethod
2 . 3 . 1 S tator 352. 3 . 2 S a lien t pole rotor 382 . 3 . 3 Round rotor 412. 3 . 4 Squirre l cag e rotor 43
2 .4 Relationship between the surface equ iva len t m odels 44and actual m ach ine quantities
2 . 4 . 1 T h ree phase w inding c u rren t and vo ltage 442. 4 . 2 S tator w inding im p ed an ce 472. 4 . 3 Squirre l cag e w inding cu rren ts and im p ed an ce 482. 4 . 4 Synchronous m ach ine ro tor w indings 50
2 .5 induction m ach ine design m ethod 54
2 .6 Synchronous m ach ine design m ethod 69
2. 7 A ppendices 87
2 . 7 . 1 List of sym bols used in the design p rocess 882 . 7. 2 M ach ine p aram eters 922 . 7 . 3 C alcu lation of the sa lien t pole facto rs Gd and Cq 106
C hapter 3 AC traction m otor design and p erfo rm an ce p red ictions 109for a high speed d iesel e le c tr ic locom otive
3 .1 introduction 110
3 .2 The traction m otor c h a ra c te ris tic and 115control strategy
3. 3 Design and p erfo rm an ce pred ictions
3 .4 Conclusions
3 .5 Appendix
3 . 5. 1 The traction m otor duty cycle
C hapter 4 The perform ance of in verter fed AC m ach in es whose phase curren t is d iscontinuous
4. 1 introduction
4 . 2 M achine m odels
4 . 2 . 1 induction m ach ine4 . 2. 2 Synchronous m ach in e
4. 3 Voltage source inverterO perating states and form ulation of system equations
4. 4 C urren t source inverterO perating states and form ulation of system equations
4. 5 C om putational p rocedure
4. 6 P erfo rm ance predictions for induction andsalien t pole synchronous trac tion m otors
4. 7 Conclusions
4 .8 A ppendices
4 . 8. 1 List of principal sym bols used in the analysis of the vo ltage and c u rren t source inverter
4 . 8 . 2 The induction m ach in e expressed in stator coord inates
C hapter 5 Experim ental verification of the cu rren t so u rce inverter m odel
5. 1 The test m ach ine and to rque m easuring system
119
138
144
145
146
147
150
150153
157
165
178
180
196
200201
203
214
215
5. 2
5 .3
C hapter 6
C hapter 7
Discussion of results
A ppendices
5. 3 . 1 D etails of test m ach in e 5 . 3 . 2 The cu rren t source inverter
Sum m ary
R eferences
220235
236 238
242
245
LIST OF SYM BO LS
ELECTRICAL AND MAGNETIC
AB
Bg
Pc
Ps
RL
Lm
P
7
a
z
q
A
KJV
E
P
Pc
PqUT
T5<p
w
Wr
CT
Re
maximum core flux density TRMS air-gap flux density T
conductor resistivity fiM
conductor surface resistance n
surface leakage inductance H
surface magnetising inductance Hphase resistance nphase leakage inductance H
phase magnetising inductance Hslot width/slot pitch coil pitch/pole pitch pole arc/pole pitch series conductors per slot slots per pole phase specific permeanceconductor current density A/mm*surface current density A/Mterminal voltage Velectric field strength V/Mconductor loss w
core loss Woutput power Wtorque Nm
torque angle degreessupply angular frequency rad/sec
angular frequency of rotor currents rad/secslip puindicates the complex real part
Final s ubscr i p t S or R d e n o t e s stator or rotor quanti ty.
7
indicates the complex conjugate e instantaneous electric field strength V/mb instantaneous magnetic flux density T
j instantaneous surface current density A/mM magneto-motive force ATI RMS phSLse current AQ total number of slotsf supply frequency Hz
f2 rotor frequency Hz
DIMENSIONS
T Slot pitchTp pole pitch
Tc coil pitch
Ta pole arcg air gap length
d slot depth
w slot width
wt tooth widthwi slot opening (semi-closed slots)Wp pole width (salient rotor)
slot conductor overhang As slot areaw^ cooling vent width
vent pitch dc stator core depthdo stator core outside diametere angle between core and end winding conductor for
diamond-ended coils
Der mean diameter of squirrel cage end ringAg^ cross-sectional area of end ring
ter thickness of end ring
Wer width of end ringmean length of coil turn
Wo core length0p half pole angle ( salient rotor )m conductor mass
iron mass A air gap surface area
CONSTANTS and FACTORS
P
k w
k p
^ x t • %xco
ksc
XendR
kendL
kpF
k g
k v
kvL
nv
K o ,x , z
6
6 i
®er
k<3
Mo
number of pole pairs winding factor
coil pitch factor
slot permeance correction factors *squirrel-cage end ring permeance correction factor
surface resistance end factorsurface inductance end factorslot packing factoriron packing factortooth flux fringing factorvent flux fringing factorvent flux fringing factor for slot leakage number of ventsconstants for core loss calculation conductor density iron densitysquirrel cage end ring densitydistribution factormagnetic constant 4rr x 10"? H/m
10
List of principal symbols used In the analysis of the voltage and cu rren t source Inverter
Induction machine
Rs
Is
Rr '
I r *
Lm
stator phase resistance stator phase leakage inductance
referred rotor phase resistance referred rotor phase leakage inductance magnetising inductance
Synchronous machine
Rs
I s
Rp •
I p .
Lmd
Cinq
stator phase resistance stator phase leakage inductance referred field winding resistance referred field winding leakage inductance direct axis magnetising inductance quadrature axis magnetising inductance
VaS'Vbs/Vcs machine phase voltages
iaS/ibS'^cS machine phase currents
i-aR' ' bR' ' i-cR' referred rotor currents(induction machine)
ip* referred field winding currents( synchronous machine )
n
H
n
H
H
H
VAA
P
<*)R
0
Tg
6«p
pole pairs
steady rotor angular velocity angular position of rotor electro-magnetic torcjue torque angle
11
rad/secdegree
Nmdegree
y phase displacement between the fundamental
component of the machine phase current and the commutation point to degree
Vg DC source voltage V
1( 0 DC link current ARdc*^3c DC link resistance and inductance D/HC inverter capacitance value FCgq equivalent capacitance value used in
inverter model F
12
C H A PTER 1
G ENERAL IN TRO D UCTIO N
The design of e lec trica l m ach ines can be broadly classified into two
distinct a re a s , nam ely , design analysis and design synthesis. Design
analysis techn iques d e te rm in e the m ach ine p erfo rm ance from a knowledge
of the m ach in e p a ram ete rs . T h ese p aram eters a re defined Initially and
rem ain unchanged throughout the execution of the m ethod. Design
synthesis . en co m p asses those m ethods in which the physical
c h a ra c te ris tic s of the m ach in e a re determ in ed from a desired perfo rm ance
sp ec ifica tio n .
The m ajority of the early problem s to be attem pted in the field of m achine
design by c o m p u te r, w ere of the design analysis type. In these cases the
co m p u ter served as an aid to the d e s ig n e r, by enabling la rg er calcu lations
using m ore a c c u ra te m ethods to be perfo rm ed . The first paper of this type 1
ap p e a re d in 1 9 5 4 . with the advent of the first m ain fram e com puters. The
co m p u ter eva luated the m ach in e p erfo rm an ce from the Initial estim ates of
the in d ep en d an t va ria b le s , and then In the light of the results the
d es ig n er in tervened and m odifed those estim ates until the required
p erfo rm an ce was ob ta in ed . The speed at which a suitable design was
found d ep en d ed to a la rg e extent upon the skill and experience of the
design e n g in e e r.
2The th em e of iterative design analysis was continued by Veinott. but with
the addition of an extra loop in which the cost effectiveness of the design
was assessed . This m ethod produced a m achine design that not only
fulfilled the p e rfo rm an ce c rite ria but also met an overall econom ic
req u irem en t. An attem pt to in tegra te the two design methods was m ade by3
the sam e a u th o r. In the a re a of sm all Induction m otor design in 1960.
As the co re p late lam inations of the sm all induction m otor were
s ta n d ard ised , the m ain in d ependant variab les were the effective turns per
phase of the w inding and the effective core length. The perform ance
U
objectives of the design could then be rea lised by a fixed step variation
of the in d ep en d an t variab les . This m ethod represen ted a partial synthesis
app ro ach to the p ro b lem , and has been sucessfully em ployed by o ther4 . 5 . 6
au th o rs . for both Induction and synchronous m achines.
P rogress in the m ore p rob lem atic a re a of design synthesis has been slow.
Only In the fie ld of tran s fo rm er d es ig n , w here the num ber of variab les
Involved Is cons iderab ly less than for e lec trica l m achines have full7 .8
d escrip tions of a design synthesis ap p ro ach been given. The ideal
m ethod of design ing any system would be by the d irect inversion of a set
of equations connecting the in d ep en d an t and dependant variab les . If this
w ere possib le the p e rfo rm an ce requ irem ents such as the output pow er,
to rq u e , speed and vo ltage could be specified and the inverted equations
evaluated to obtain the n ecessary m ach in e d im ensions. In practice the
equations would be underdefined as th ere a re m any m ore dim ensions to
be solved fo r. than there a re Input specifications. The problem can be
eased slightly by adding an overall req u irem en t that the solution obtained
m ust re p re s e n t a m inim um cost or m inim um w eight design.
9As an ap p ro ach to the d irec t m ethod of solution M Iddendorf. derives an
equation re la ting the p erfo rm an ce ch arac te ris tics of an induction m otor to
the ro tor s ize . A lthough , whilst this re lationship considers such quantities
as the pull out to rq u e , starting torque and cu rren t. It does not Include
w hat m ust be one of the m ost Im portant perfo rm ance c rite ria , the power
fac to r.
As a d irec t ana ly tica l inversion of the design equations is not possible, an
iterative app ro ach to the prob lem has to be adopted. However as the
design of e le c tr ic a l m ach in es contains a high d eg ree of d iscreteness, any
iterative tech n iq u e m ust be ab le to work within this lim itation. Typical
d iscre te variab les a re the s tandard conductor and slot sizes, num bers of
15
slots and co n d u cto rs , which m ust have an In teg er value, and fram e size .
for which a lim ited num ber m ay have to cover a wide range of output10
pow ers. A paper p resented by C h a lm ers and B ennington. describes a
m ethod for the design of la rg e squ irre l c a g e Induction m ach ines. In which
both design analysis and design synthesis m ethods are used In
conjunction with Iterative tech n iq u es . to produce an econom ical
c o n verg en ce sch em e .
D espite the ab s e n c e of progress in the fie ld of design synthesis, work has
continued in the a llied th eo re tica l study of design optim isation.11.12
d isreg ard in g fixed sizes and d iscontinu ities .
In the following w ork, a g en era l design m ethod Is presented that Is
suitab le for the design of induction and s lip -r in g synchronous m achines of
the round ro tor o r sa lien t pole type. The design techniques presented in
C h ap ter 2 fall b roadly Into the c lass of design analysis , but retain som e
e lem en ts of a design synthesis ap p ro ach .
The m ethod Is based upon m ach ine m odels that em ploy surface
q uantities . This type of m odel displays c le a rly the In teraction between flux
and c u rre n t, and Is read ily ad ap tab le for design use as the am ount of
detail req u ired is kept to a m in im um . The geom etrica l and e lec trica l
c h a rac te ris tics of the m ach in e a re evaluated with the m inim um of
com putational e ffo rt and without the need to specify a large num ber of
input va riab les . This obviously requ ires som e reduction of the m ach ine
p a ra m e te rs , if the design process is to rem ain as concise as possible.
To this end the d im ensions of the m ag n etic c ircu it and the form ulae for
the ca lcu la tion of the m ach ine res is tan ce s , inductances and m asses have
been s im p lified , but only w here it is fe lt to be benefic ia l In the a re a of
reducing the n u m b er of input variab les to be specified .
16
T h e search for an optim um design is not cons idered In this work, but as
the s im plified m ach in e design equations use the m inim um num ber of
v a ria b le s , they a re In a form that is read ily adap tab le to som e sort of
sensitivity analysis .
The DC m otor, in traction ap p lica tio n s , has been developed over recen t
years to a high level of soph istication . This m otor has an excellent
overload cap ab ility . Im m unity to line voltage varia tions, and provides good
torque sharing w hen driving w heels of d iffe ren t d iam eters . In the past 20
years th e specific output has been in creased by approxim ately 70 % . this
in c re a s e being la rg e ly due to the use of Im proved insulating m ateria ls and13
h ig h er working tem p era tu res . In c re a s e s in m otor output how ever, a re
lim ited by co nstra in ts on the com m utato r p erfo rm ance. Removal of the
co m m u ta to r rem oves the lim it on high speed o peration , and assum ing
equal cu rren t and flux d en s ities , s ign ificantly shortens the length of the
m a c h in e , and gives a c o n s id e rab le reduction In w eight. A reduction In
m ach in e length and w eight en ab ies m ech an ica l changes to be m ade that
are advan tageous in traction app lica tions . For exam ple a sm aller
g earw h ee l d ia m e te r may be used, due to the reduced g ear centre
d is ta n c e , enab ling the veh ic le height to be low ered. Any weight reduction
will contribu te to im proving the ride quality of the bogie due to a reduced
track load ing . A lternative ly it will m ean that h igher operational speeds a re
possib le w ithout in creas in g the track load ing .
The railw ay en v iro m en t is harsh for bogie m ounted com ponents. The steel
w heel - rail system im poses high v ibrational fo rces , and dynam ic loading
of the traction m otor can reach 50 g. All these factors ren d er the DC
m otor co m m u tato r and b ru sh g ear vu lnerab le to w ear. This increases the
freq u en cy at which they must be in sp ected , with a consequent in crease in
m ain ten an ce costs.
17
The use of AC m o to r-In v e rie r drives in traction applications has grown14. 15. 16. 17. 18. 1 9 .2 0
s tead ily , and results of som e European system s
have been rep o rted . Early AC e lec trifica tio n m ade use of 3 -p h a s e
Induction m ach in es , but only as substantially fixed speed m achines with
pole c h a n g e w ind ings. This approach suited heavy locom otive work and
provided useful reg en era tive b rak ing , but for light weight high speed
ap p lic a tio n s , the g re a te r flexibility o ffered by a variab le speed drive Is
req u ired .
The w ork of C h ap ter 3 considers a specific app lication of the design
m ethod . D esigns are presented for induction and slip ring synchronous
m ach in es that satisfy the requ irem ents of a traction m otor for use in a
high speed d iesel e le c tric locom otive. The proposed designs represent an
a lte rn a tive to the DC traction m otor and m echan ica l transm ission in
presen t use. The AC traction m otor is in tended to be mounted between
the w h ee lse t and provide a variab le speed drive through a reduction
gearbox.
As the re lative sizes of the m ach ine and pow er supplies a re very much
d ep en d an t on the way In which the system is con tro lled , two com m only
used sch em es a re c o n s id ered . The perfo rm an ce of the resulting designs
is co m p ared on a sinusoidal basis and th e ir suitability for this particu lar
traction app lication Is assessed .
As the traction m otor designs of C hap ter 3 a re based upon sinusoidal
supply c o n s id e ra tio n s , th e re Is a need to determ ine the ir perfo rm ance
w hen being supplied with an inverter of the appropria te type. In C hapter
4 co m p u ter m odels a re p resen ted , that enab le the steady state
p erfo rm an ce of Induction and round ro tor or salient pole m achines to be
p re d ic te d , whilst being fed from e ith e r a voltage or curren t source
In verte r, o pera ting In the 120 d eg ree conduction m ode.
18
In fo rc e com m utated Inverters the thyristor conduction periods a re well
d e fin e d , and can ran g e from fractions of a d eg ree In pulse width
m odulated Inverters , to 180 d eg rees of the output period in square wave
Inverte rs . For a voltage source In verter operating In the 180 d eg ree
conduction m ode, th ree thyristors a re gated on at any Instant. The
com m utation of one thyristor and the firing of its com plem entary thyristor
in the sam e leg o ccu r a t the sam e tim e . This produces a precisely
defined output voltage waveform and en su res continuous phase curren ts .
The g a te firing pulses have to be app lied alm ost sim ultaneously to top and
bottom leg thyristors m aking the logic design relatively com plex. Several
tech n iq u es have been presen ted for the analysis of the 180 deg ree square
wave In v e rte r, with the assum ption that the m otor Is supplied from a2 1 . 2 2 . 2 3
known vo ltage w aveform .
For vo ltage source inverters operating In the 120 d eg ree conduction mode
the in verte r thyristors a re perm itted to conduct for 120 degrees of the
output perio d . As oniy two devices a re conducting at the sam e tim e, a 60
d e g re e gap exists betw een the com m utation of one thyristor and the
turn ing on of its co m p lem en tary device in the sam e leg. This considerab ly
sim plifies the g ate pulse in fo rm atio n , but can lead to discontinuous phase
cu rren ts in high pow er fac to r loads, due to the disconnection of one of
the m ach in e phases during this period . During the period for which the
phase cu rre n t is zero the m ach in e back em f appears at the output
te rm in a ls of the inverter. T h e m agnitude and duration of this voltage is a
function of the load . Owing to the varying topology of the Inverter c ircu it.24
the analysis is not s tra igh tfo rw ard . Lipo and Turnbull have developed a
m ethod of utilising state variab le analysis for the prediction of steady state
w aveform s for 120 and 180 d eg ree Inverters feeding constant speed low
pow er Induction m ach in es . A tensor m ethod presented by A l-N lm m a and 25
W illiam s investigates the tran s ien t and steady state perform ance
c h a ra c te ris tic s of an induction m otor for both 120 and 180 d eg ree Inverter
19
conduction m odes.26
Lockwood utiilised an an a lo g u e co m p u ter m odel which again predicted
the tra n s ie n t and steady state p e rfo rm a n c e , but for a much la rg er 200KVA
traction system using tubular axle Induction m otors.
27The c o n c e p t of the curren t so u rce inverter was described by W ard in
1964 . In the cu rre n t source in verte r, the dc link curren t Is held constant
and by sw itching the Inverter thyristors at the requ ired rate the m achine
speed Is continuously va riab le . As one of the m ain operating
c h a ra c te r is tic s of the cu rren t source inverter Is the generation of voltage
spikes of severa l tim es the load term ina l vo ltage , the application of such
Inverters to Induction m otor drives had to w ait for the developm ent of high
voltage thyristors . The first g en era l co m m erc ia l description of an Induction28
m otor d rive using the cu rren t source inverter was given by Phillips. in
1972. If com m utation of the m ach in e cu rren t is considered to be
instan taneous the c u rren t waveform is re c ta n g u la r. In 120 degree blocks.
The analys is of the Induction m otor operating on q u as l-sq u are currents Is2 3 ,2 9 30
d iscussed In d e ta il. and fo r the synchronous m otor. In practice
the com m utation of cu rren t in the Inverter Is not instantaneous. During
co m m u ta tio n , the m ach ine res is tan ce and Inductance form a part of the
com m utation c irc u it together with the com m utation cap ac ito r, and hence
the varia tion of c u rre n t follows a dam ped sinusoid.
A m ore deta iled analysis of the com m utation process and a derivation of
a se ries eq u iva len t c ircu it for the induction m otor was presented by31
F a rre r and M Iskln. How ever approxim ations w ere m ade In their work by
neg lectin g the stator res is tan ce and assum ing the m achine back em f to be
constan t during the com m utation In terval.
An exact m odel of the c u rren t source Inverter Is presented In C hapter 4
which a c cu ra te ly re flec ts the way In which a curren t source would be
20
derived in p rac tice . No assum ptions a re m ade about the m ach ine back
em f during com m utation , and the effect of the DC link Inductance Is
taken Into acco u n t by con s id erin g the Inverter to be fed from an ideal
voltage s o u rce .
B ecause of the in h eren t sw itching sym m etry in both the voltage and
cu rren t so u rce Inverters , it is necessary to consider only one sixth of a
cycle of in verte r op era tio n . A com ple te solution may then constructed from
the steady state 60 d e g re e values.
The p e rfo rm a n c e equations fo r the Induction m ach ine a re derived In term s
of the a c tu a l m ach in e vo ltages and curren ts without resorting to a two axis
tran s fo rm atio n . All the Induction m ach ine variab les a re re lated to a
c o o rd in a te system that is fixed in the stator, and thus the dependency of
the m atrix coeffic ien ts on the rotor position Is rem oved. This reduces the
am ount of com puting effort requ ired to obtain a steady state solution, as
the s ta to r and rotor cu rren ts a re at the sam e frequency , and enab les the
Inverter - m ach in e equations to be set up with relative ease .
P e rfo rm a n c e pred ictions a re presented for w hat is fe lt to be the two most
su itab le designs for use In a high speed passenger train application .
C u rren t, vo ltage and torque waveform s a re g iven, and a com parison is
m ade betw een the harm on ic torques p resen t for each m o to r-in verte r
com bination at norm al operating levels.
In co n c lu s io n , experim enta l results a re presented In C hapter 5 to verify
the c u rre n t source m odel. A 5KVA laboratory squirre l cage induction
m ach in e is used for this purpose. The stator fram e of this m ach ine is
iso lated from the rotor assem bly and is supported on a force m easuring
platfo rm . This enab les dynam ic m easurem ents of the rotor shaft torque
pulsations to be reco rd ed . Inverter voltage and curren t waveform s a re also
shown.
21
CHAPTER 2
THE D ESIG N OF IN D U C TIO N A N D SYNCHRONOUS MACHINES
BASED ON A SURFACE EQUIVALENT MODEL
2 . 1 In troduction
T h0 g en era l m ach in e design techniques presented in this chap ter are
a p p licab le to s q u irre l-c a g e induction m ach in es , and s lip -rin g synchronous
m ach ines of the ro u n d -ro to r or salient pole type.
The design m ethod is based upon m ach ine m odels that em ploy surface
q u an tities '. Th is type of m odel has the advantage of displaying c learly the
in teraction betw een flux and c u rren t, whilst enab ling the am ount of detail to
be kept to a m in im um . Maxim um perm issib le values of stator and rotor
conductor cu rren t density and core flux density a re specified as inputs to the
design p ro cess , th e ir values being chosen accord ing to heat dissipation and
saturation levels. The m ach ine w indings are represen ted by thin sheets of
conductors on the a ir gap surfaces of the stator and rotor m em bers.
M ach in e res is tan ce and leakage inductance effects are incorporated into the
m odel by giving the winding conductor sheets suitable values of surface
res is tan ce or inductance . The induction m ach ine may then be readily
rep resen ted by a conventional equivalent c ircu it in which surface quantities
are used.
As the resu ltant a ir gap flux density of the synchronous m achine Is due to
sep ara te stator and rotor com ponents , the re lationship between current and
flux density In this case Is m ore easily d isplayed on space and tim e phasor
d iag ram s.
In view of the la rge num ber of d im ensions and other variab les Involved, som e
reduction of the m ach ine form ulae Is necessary If the design process Is to
rem ain as concise as possible. To this end the dim ensions of the m agnetic
c ircu it and the m ach ine res is tances . Inductances and m asses have been
sim plified w herever possible. In each case an exact expression Is derived
23
and then with the use of appropria te sim piifying assum ptions, a reduced form
is obtained that is m ore su itab ie for use in the design process.
It wiii be shown that for a given pole num ber, and assum ing an equal flux
density in ail parts of the m agnetic c ircu it, the stator c ro ss -sec tio n can be
com plete ly defined by the cho ice of th ree variab les . These are the outside
d ia m e te r, the slot dep th , and the ratio of slot width to slot pitch.
Having defined a su itab ie stator c ro s s -s e c tio n , consideration may then be
m ade of the m axim um stator surface cu rren t loadings atta inab le by that
c ro s s -s e c tio n , for a specified conductor cu rren t density.
in the case of the induction m ach in e , the a ir -g a p length is specified as an
Input to the design process and thus enab les the rotor geom etry to be
defined w hen the stator c ro ss -sec tio n is known. The perfo rm ance of the
induction m ach in e design is then d e te rm in ed , for a given core length , by the
m axim um perm iss ib le stator and rotor surface curren t loadings for an
app ro p ria te m ach in e geom etry.
For the synchronous m ach in e , the stator c ro ss -sec tio n is defined in the
sam e way as the induction m ach in e , for m axim um perm issib le current and
flux loadings. An iterative process is then adopted to find , in itiaiiy. an air
gap length that gives a stator com ponent of a ir gap flux density that is equal
to or below the m axim um air gap flux density a llow ab le , i. e. no rotor
com ponent of a ir gap flux. The a ir gap length is then increased in steps
until the condition is reach ed for which th ere is insufficient rotor conductor
a re a rem a in in g , to support the required value of rotor surface current
load ing , at a specified m axim um rotor conductor density, to set up the
requ ired rotor com ponent of a ir gap flux.
2U
In o rd er to dem onstra te the g en era l design m ethod and to form a bridge
betw een this and the next c h ap te r, one design for each m achine type is
shown. T h e s e m ach in e designs a re rated at traction levels and satisfy the
req u irem en ts of a m otor for use in a high speed d iesel e lec tric locom otive.
The full sp ec ifica tio n of a m ach ine for use in this app lication is described in
m ore deta il in C hap ter 3.
25
2. 2 M ach in e m odels and equivalent circuits
2 . 2 . 1 T h e Induction m ach ine
The m odel of a tw o -p o le induction m ach ine with a uniform air gap is shown
in cross section in Fig. 2 . 1 . The stator and rotor are represented by
unslotted m em b ers of a hom ogeneous m ateria l that has a high value of both
resistivity and p erm eab ility . The distribution of the conductors around the
cy lindrica l su rfaces is such that when the norm al phase currents are flow ing,
only the fu ndam enta l com ponent of the m mf produced by the actual winding
is g iven. The effect of res is tance and leakage inductance are m odelled by
giving the w inding conductor sheets suitab ie values of surface resistance and
inductance .
As the m odel of F ig . 2. 1 is assum ed to be a representative section of the
actual m a c h in e , a tw o -d im en s io n a l analysis is appropria te . A relationship
betw een the a irg ap flux density and the w inding surface currents can be
obtained by reca llin g A m p ere 's Law in the fo rm .
[ Hdfi = f Jds (2.1)
For the path shown in Fig . 2 . 1 . assum ing only rad ial gap flux and an
a ir -g a p length that is sm all in com parison to its m ean radius; and - co. p
dbgi s " Î R ( 2 . 2 )
Mo ay
If each of the variab les in equation ( 2 . 2 ) is rep resen ted by a num ber of the
form
a = Re]
/2 Ae
rryo)t - —
Tpj
26
s t a t o r S u r f o c » C u r r e n t
D ensi ty
R o t o r S u r f a c e C u r r e n t
D e n s i t y
a i r g a p g
bg * 3 b g dy
îÿ"bg
bg
Fig 2.1 Machine Surface Model
:s R
%Eg 1
%
Fig 2 .2 Surface Equivalent C i r c u i t fo r A i r Gap
E a
s PssO ^ 'V V W '-
' s s-cmJLr-
E g = B g / ° S Rcr
Fig 2 .3 Induc t i on Machine Sur face Equivalent Circuit
27
w here A is com plex, then
grr- j Bg = Jg - JR ( 2 . 3 )
The voltage Induced in the m ach ine windings Is proportional to the electric
field strength at the conductor surface.
The relationship between the e lectr ic field strength and the air gap flux
density may be obtained by applying F araday 's Law around a path In the
plane of the air gap I. e.
ae abg
ay a t
or In complex RMS term s
rrEg
Tt — w Bg ( 2 . 4 )
Com bining equations ( 2 . 3 ) and ( 2 . 4 ) gives
EgJs - JR - - iw im ( 2 - 5 )
Where f ^ is the surface m agnetising Inductance of the air gap, and Is given
by
(m = ( 2 . 6 )gTT
The surface equivalent circuit representing equation ( 2 . 5 ) Is shown in Fig
2 . 2 .
28
On the stator winding conductor sheet the resultant e lectric field strength is
m ad e up of two separate com ponents, a com ponent due to the air gap, and
a co m ponent that results from the stator current sheet Im pedance drop,
thus the e lec tr ic field strength at the stator term inals is
'a “ * s ( Pss iw*ss) ^ Eg (2.7)
w here Pss and i^ s represent the surface resistance and leakage inductance,
respective ly , of the stator winding. On the rotor winding the surface current
Is c ircu lated by the a i r -g a p electric field. 1 he a ir -g a p electric field Is
g en era ted by the air gap f lux-wave, whose speed with respect to the rotor Is
given by
W — = (JO) rads/sec
Thus on the rotor the e lectric field strength Is given by
TTEf= aw Bf ( 2 . 8 )
C om paring equations ( 2 . 4 ) and ( 2 . 8 ) gives E' = aE, so the rotor equation
correspond ing to equation ( 2 . 7 ) for the stator is:
Eg = aEg = (PSR + jewggR) Jr
orPSR
iw *5R jR (2.9)
w here psR and fisR represent the surface resistance and leakage inductance
respectively of the rotor winding. Combining equations ( 2 . 5 ) , ( 2 . 7 ) and
( 2 . 9 ) leads to the Induction m achine "surface equivalent circuit" of F ig .
2 . 3
29
2. 2. 2 The synchronous m achine
The round rotor synchronous m ach ine model is similar to that of the
Induction m ach ine given In Fig. 2 . 3 , but In this case operation Is at
synchronous speed. In the synchronous m ach ine the rotor currents are not
driven by the a ir gap e lectr ic field. E g , but a re defined by a DC coll system.
In the m odel of Fig. 2 . 1 , the direction of rotor currents are assumed to be
reversed and leads to the equivalent circuit of Fig. 2 . 4 . As the resultant air
gap flux Is due to two Independent com ponents . It Is more convenient to
display the re lationships between current density and flux density on a space
and time phasor d iag ram . Fig. 2 . 5 .
The e lec tr ic field strength due to the stator surface current density, Jg, Is
given by
coTpBgEg = ~ — Js ~ ( 2 . 1 0 a )
a nd the e l e c t r i c f ield str eng th due to the rotor sur fa ce c ur ren t densi ty . J r .
Is
WTpBR ^<^TpEr = - = ^2g (2.10b)
The m ach in e output power Is
Pout = ErJ s a s in 6<r ( 2 . 1 1 )
w here S j Is the torque angle and A the air gap surface area .
30
E a
Fig 2 . 4 Round Rotor Synchronous Machine Su r f a c e Equivalent C i rc u i t
Jc
BgBr Ea
Fig 2 .5 Round Rotor Synchronous Machine Phasor Diagram
31
In synchronous m ach ines of the salient pole type the resultant flux distribution
does not co incide with that of the resultant mmf of the windings. The air
gap flux produced by a distributed mmf depends upon the orientation of the
mmf axis with respect to the sallency and the gap re luctance presented to
It.
The effect of a non uniform air gap length Is represented by giving the stator
flux sep ara te com ponents along each of the two axes of symmetry of the field
structure. These com ponents are related to the flux produced In a uniform
air gap m ach in e by the factors C(j and Cq. and a re described In Appendix
2. 7. 3. The phasor d iagram representation of the salient pole synchronous
m a c h in e . In surface term s. Is shown In Fig. 2 . 6 .
The d and q axis com ponents of the stator surface current density, from Fig.
2 . 6 . a re
J q — J g C O S 5 * p
and
»Jq — J g sin 6<p
The stator flux and e lec tr ic field strength may be similarly resolved Into two
com ponents a long the two axes of symmetry. For the stator flux. Bg
(-df^D^pTdBa = „g <2 12= )
and
Cq^TpiJqBq = - ■ (2.12b)
32
Bg
E d
fcc ?dojtSS
Fig 2 .6 S a l ie n t Pole Synchronous Machine Phasor Diagram
33
and for the e lectr ic field strength. Eg
Ed = IT n^g(2.13a)
and
CO TpBq CqW/iQTpJq- w4mqJq
n^g(2.13b)
For the rotor, which is a ligned with the d axis, the electric field strength E r
due to the surface curren t density Jr . is
Er =TT g
The output power, in surface term s, is then
(2.14)
Pout -co«7<
ErJs sin 5t + ((md " ^mq) sin 26^ A (2.15)
The first term in equation (2 . 15) represents the excitation torque, and the
second term the re luctance torque due to the saiiency of the rotor.
36
2 . 3 A consideration of the geom etric and m agnetic circuit aspects
of the design method
2 . 3 . 1 Stator
The design of any AC m achine is Initiaiiy concern ed with the choice of a
suitabie stator. The most critical part of the stator magnetic circuit. Fig.
2 . 7 . is likely to be the teeth. The tooth flux density is related to the gap
density by the ratio of the m inimum tooth width to the slot pitch. H ence , for
any limiting value of fiux density in the teeth, there is a corresponding
maxim um perm iss ib le density in the a i r -g a p . which Is determ ined by the ratio32Of slot width to tooth width The maximum allowable gap density. B g . in
term s of the ratio of slot width to slot pitch, p. assuming rectangular slotsA
and a maximum core fiux density of B. Is given by
BBg = (1-/3) (2.16)
✓ 2
where Bq is in RMS terms and p ■= —
The total pole fiux divides into two equal com ponents in the core to pass into
the ad jacen t poles. in order to acco m m odate this fiux at a density of B.
T es ia the backing core depth must be given by the following expression
Bdc = - ^2 Bg — (2.17)
C om bin ing equations ( 2 . 1 6 ) and ( 2 . 1 7 ) gives an expression for the core
depth required- in terms of the poie pitch and the ratio p.
'P— (1-/3) (2.18)TT
35
Be
Ts
ds
Bg
Be - core d e n s i t y Bt - t o o th d e n s i t y Bg - gap d e n s i t y
Fig 2.7 F lux D e n s i t i e s in the S t a t o r
RM
Fig 2.8 Machine Cross Sect ion (squ i r re l cage or round rotor )
36
With the depth of backing core required given by equation ( 2 . 1 8 ) , the pole
pitch, for a specified slot depth , dg, and core d iam eter, dg, may be
obtained from
- - - dc - ds2prp
2 tt
On com bin ing this expression with equation ( 2 . 1 8 ) the pole pitch is given
by
TT(do - 2dg)
2[p + ( 1 - /3)] (2.19)
The stator Iron cross section. Fig. 2 , 8 , is then uniquely defined for a
m ach in e having p pole pairs, by only three variab les , the outside d iam eter,
dg, and slot depth , dg, and the ratio of slot width to slot pitch, p.
The total stator iron m ass may now be determ ined for any core length, Wg,
by use of the following expression
■ doz
do 2TT 2 ~
- dc - ds &P9sWsds
This equation may be simplified by assuming that there are no cooling vents
to give
do2
do 2TT 2 ■
— - dg - ds - 2p/3Tpdg ( 2 . 20 )
w h ere kjg is the stator iron packing factor.
At this stage it is appropria te to introduce an expression for the core loss,
as this loss is proportional to the mass of the stator iron when the flux
37
density and frequency are known. The expression used is
Pg = Kgf^B^is Watts (2.21)
For a typical e lec tr ica l steel in 0. 5 mm laminations the constants in equation
( 2 . 2 1 ) have the following values:
K g = 0 . 0 2
X = 1 . 1 3 6
z = 1 . 7 6
These values a re assum ed throughout this work.
2 . 3 . 2 Salient pole rotor
Two types of lam inated rotors are considered , the round rotor type and the
salient pole type. The DC field winding in each case is fed by slip rings.
The geom etry of the salient pole structure is potentially the more difficult to
d escribe accurate ly . To simplify the structure somewhat, a simple
c ro s s -s h a p e d one p iece lamination has been adopted. Fig. 2. 10. which
does not include the provision for dam per windings. Any discrepancy
between the sim ple iron circuit proposed and a production stamping may be
acco m m o d ated by the use of an Iron packing factor. k|p. for the calculation
of rotor m ass. S im ilarly any d iscrepancy In the slot area available for the
field winding may be taken ca re of by use of the slot packing factor, kppp.
The geom etry of o n e -h a l f of a salient pole rotor Is shown In F ig . 2 , 9 . If the
rotor poles are assum ed to be rectangular then the maximum Interpolar depth
Is:
W p
âRM ' Ro - 2 sin 0p (2 22 )
38
slotRM
pole
0
Fig 2.9 Rotor S lo t Dimensions
Fig 2.10 Sal ient Pole Ro to r
39
where 0p = — and Rq is the rotor radius, and is given by
doRo = - dc - dg - g (2.22a)
In o rder to acco m m o d ate the stator flux at the sam e flux density, the pole
width. Wp. must be equal to twice the depth of the stator backing core.
An approxim ate value for the a rea of the conductor filled part of the slot, if
the slot is filled to a depth dp . Fig. 2 . 9 . with conductors, can be obtained
from
one half the slot area = (area of sector OCD)- (area of sector GAB)
- (area of rectangle ACFE)
1 2 2 d%wpi . e . - As r - Ro - (Ro - dR) - — —
or Asr = ©pdR ( 2Rq - d^) - Wpdp (2.23)
The m axim um availab le slot area for the accom m odation of the rotor winding
is there fore
A g R M * G p d R M ( 2Ro ~ d p ^ ) “ ^ p d R M (2.24)
and the correspo nd ing pole area is
Ap - 6pR^ - AsRM (2.25)
The rotor iron mass for any core length. Wq, is then
( " iR - ( ^ c ~ MyWy) 2p 6 2 A p
40
If the rotor is assum ed to have no vents this equation may be simplified to
ïûiR — 2p l^iR 62 Ap ( 2 . 2 6 )
2 . 3 . 3 Round rotor
F orm ulae for the slot d im ensions of the round rotor are developed from those
given in the previous section for the salient pole case. If the same diagram
is used for the round rotor. Fig. 2 . 9 . as for the salient poie rotor, for each
slot pitch then
wt + Wr®P “ 2Ro
where w p is the width of the slot opening, and w is the tooth width.
Thus the maximum slot depth is given by
dRM = Ro - + WR (2 2 7 )
If the rotor slot and tooth width are assum ed to be equal to that of the
stator, then equation ( 2 . 2 7 ) may be simplified to
dRM = Ro/3 ( 2 . 2 8 )
As the slot opening is usually sm all, when com pared to the rotor
c ircu m feren c e , the slot cross section is considered to be tr iangular. F ig .
2. 8. This gives a maximum available slot area
<3rMWr
Asrm Ô
If the slot is filled to a depth dp with conductors, then the rotor slot area will
be.
41
As r = WRdR ZdRM(2.29)
This equation may be abbreviated by defining an effective slot depth,
dR* “ dpdR
1 - -
ZdRM(2.30)
Having defined the slot a re a , the rotor iron cross section and hence the iron
mass may be ca lcu lated
m i R = G i k i R (T T R o - q R P d R M W R ) ( W q , - H v W v ) (2.31)
A simplified form of equation ( 2 . 3 1 ) may be obtained by making the following
assumptions:
i) the rotor has no vents ii ) the stator and rotor have an equal number of slots
per pole i.e. qR = 3qg ill) the stator and rotor slot openings are equal
i.e. Wp = ws This gives an iron mass of:
miR * WckiR6iRo(TrRo - PTppZ) (2.32)
42
2. 3. 4 Squirre l cag e rotor
The rotor of the induction m ach ine is of a conventional squirrel cage
construction. It was felt that a choice of rotor bar shapes would unduly
com plica te the m ethod. T here fo re a sim ple V shaped slot was adopted,
sim ilar to that used for the round rotor synchronous m ach ine . Figs. 2 . 8 and
2. 9.
If It is ag a in assum ed that the rotor slot width and pitch are equal to those of
the stator, the maximum slot depth availab le for parallel teeth is
dRM = Ro/3 (2.33)
The rotor core rad ius . Rq . is expressed in term s of the stator dimensions for
a specified a ir gap length.
The rotor iron cross section is ca lculated in the sam e way as that leading to
equation ( 2 . 3 1 ) . but with three times as m any slots, i . e .
m iR = G ik ip ( ttR^ - 3qR pdR j^R )(W c - n^Wy) ( 2 . 3 4 )
A simplified form of this equation may be obtained by assuming that
i) the rotor has no vents
i i ) 9R = 95
iii) w r = w g
This gives
m^R = 6ikiRRo(TTRo - pTp/3^)Wc ( 2 . 3 5 )
43
2. 4 R e l a t io n s h ip b e t w e e n the s u r fa c e equ iv a le n t models and
ac tu a l m a c h i n e quant i t ies
2 . 4 . 1 T h re e phase winding current and voltage
The re lationship between the actual winding phase current and a surface
current distribution may be obtained from a consideration of mmf.
Application of A m p e re 's law to a uniform air gap (section 2 . 2 ) shows that
the mmf can be cons idered to be due to a sinusoidal distribution of surface
cu rren t, i. e.
tt
Js ■= — Ms (2.36)
C onsidering only one phase of a balanced winding, the peak value of the
fundam enta l com p onent of m mf. in terms of the phase current, is given by:
A ^ Zgqsk^s AMs = - ---2--- Is (2.37)
Equations ( 2 . 3 6 ) and ( 2 . 3 7 ) may be com bined to give an expression for the
surface curren t density in terms of the actual phase current.
3zgqskws Is (2.38)
w here Jg and Ig a re RMS quantit ies, and a factor of 3 / 2 has been introduced
to account for the other two phases.
The surface cu rren t of equations ( 2 . 3 6 ) and ( 2 . 3 7 ) may be thought of as
being due to a current flowing in a thin sheet of sinusoidally distributed
conductors.
44
For o n e phase
No Is-^s
Zzgqgkwswhere Ng = c o n d u c to rs /m e tre
Having defined a conductor distribution, the induced phase voltage may be
obtained by integrating the a i r -g a p e lectr ic field strength distribution over the
conductors .
If the field strength at the winding surface is:
ea = Ea cosTTy
wt - --T p .
then the voltage induced in p p o le -p a irs is
r^P-Tp A= Wc J E a cos
rrywt - —
^P
TryNo cos — dy7p
T h e RMS induced voltage per phase is then
Vg *= 2p WcZgqgkwgEa (2.39)
A re lationship between the conductor current density and the surface current
density is obtained from the winding current. If one stator slot Is
co n s id ered , the conductor current density is. using RMS quantities
Z g is
w h ere kpps is the slot packing factor. This factor takes account of the
reduction in slot a re a caused by the need for conductor insulation and a slot
w ed ge .
45
The surface curtent density, from equations ( 2 . 3 8 ) and ( 2 . 4 0 ) is
Kg 3qg kyyg kppg Agg
This expression may be simplified by assuming:
i) a rectangular stator slot whose area, Agg » Wgdgii) that each pole has 3qg slots of pitch Tg
i.e. 3<3gTg ~ Tpiii ) a distribution factor of 1
This gives
Js - Kg kppg kpg dg (2 .41)
In the design process a limiting value is assigned to the stator conductor
curren t density. This value determ ines the maximum permissible current
loading when the stator geom etry is known.
46
2. 4. 2 Stator winding im pedance
The norm al induction and synchronous m achine equivalent circuit
p aram ete rs , including expressions for conductor mass, are derived in
Appendix ( 2 . 7 . 2 ) . Most of the form ulae presented there are based upon
those given in re fe ren ce s 33 and 34 . Following each derivation a simplified
form is g iven, and it is this reduced from that is used in the following
determ ination of the surface equivalent im pedances .
The re lationship between the actual winding im pedances and their equivalent
surface quantities is obtained by equating VA. For a uniform air gap
m ach ine with no rotor conductors:
3Is [Rs ^n)] “ ZpTpW^Jg [Pss jw(*ss "*■ *m)]( 2 . 4 2 )
Substituting the winding cu rren t. Ig. from equation ( 2 . 3 8 ) into the above
expression and equating terms gives the following :
stator surface resistance pgg = Rg KgsuRF stator surface leakage inductance fgg = Lg Kggupp
surface magnetising inductance fm = KgguRp
Tpwhere the stator surface constant Kggupp =G( Zgqgkwg )^pW(3
The surface equivalent im pedances In terms of the machine dimensions and
constants, using the simplified version of the m achine param eters from
Appendix ( 2 . 7 . 2 ) . are as follows:
47
Pcs KendRs
Kps KpFS dg/3Pss = — -------------------- ( 2 . 4 3 )
F*o Kxco ^s KendLs«SS - ; -------------- ( 2 . 4 4 )
3/3 kpg
Mo Tptin - — 7 — ( 2 . 4 5 )
n g
w here the factor kendpg and kend|_s are term s associated with the end
connections of the w inding, and are described in Appendix ( 2 . 7 . 2 ) ,
2. 4. 3 Squirrel cage winding currents and im pedance
The squirrel cage rotor winding is equivalent to a balanced three phase
winding with one conductor per slot. The conductor currents are sinusoidally
distributed, as they are driven by a sinusoidal air gap field. A sinusoidal
distribution of winding currents implies a winding factor of unity. Thus the
rotor surface current density from equation ( 2 . 3 8 ) is.
39R I rJr = — ------ ( 2 . 4 6 )
or in term s of the conductor curren t density
Kr 3qR KpFR AgRJr ^ ;
Tp
With the rotor slot defined as in section 2. 3. 4 this may be simplified to give
J r K r k p F R /3 d R . ( 2 . 4 7 )
48
As all the avai lable slot a rea is used, for all induction machine designs
presented in this v/ork. the effective slot depth dp' from equation ( 2 . 3 0 ) is
d R w / 2 A d irect relationship now exists between the stator and rotor surface
current loadings of the model and their respective conductor current densities
(eq u a tio n s ( 2 . 4 1 ) and ( 2 . 4 7 ) ) . It is this relationship, via the m achine
dim ensions and constants , for a given maximum core flux density, that sets
the p e rfo rm a n c e limits for a particular induction m achine design.
The rotor equivalent surface im pedances may again be found by equating VA
i . e . :
3 I r ( R r + ] W L R ) = Z p T p W ^ J R ( P S R + j w * S R ) ( 2 . 4 8 )
Substituting for the winding curent. 1r . from equation ( 2 . 4 6 ) gives the
following
rotor surface resistance psr = Krsurj? Rr
rotor surface leakage inductance ÉsR “ Krscjrf &R
Tpwhere the rotor surface constant Krsurf -6qR pwc
Using the expressions for the rotor res is tance and inductance from Appendix
( 2 . 7 . 2 ) . the rotor surface quantities may be obtained in terms of the
m achine d im ens ions , i . e .
PCR ^^endRR= kpFR dp. p (= 49)
Mo <3r kendLR«SR = ( 2 . 5 0 )
49
2 . 4 . 4 Synchronous m achine rotor windings
a) Round rotor
The rotor winding of a round rotor synchronous m achine is equivalent to one
phase of a th ree phase w inding, and therefore an expression for the surface
curren t density in term s of the actual winding current can be obtained from
equation (2 . 38 ) .
/ 2 ZRqRk^RlRJr = (2.51)
or in te rm s of the rotor conductor current density, Kr ,
V2 qRkv,R)CpFRAsRKRJr =
A simplified form of the above expression can be obtained by making the
following assum ptions:
i) the coil sides of the winding are spread over one pole pitch.2
This gives a winding factor of k^R = —ii ) the stator and rotor have an equal number of slots
i.e. SqgTg = qpTR = Tpiii) the rotor slot width and pitch are equal to those of the statoriv) the rotor slot geometry is as defined in section (2.3.3),
with an effective depth, dR*.The rotor surface current density now becomes
2/2 kpFR p dR* Kr Jr = ----------------- (2.52)
The equivalent surface resistance of the rotor winding is obtained by equating
the actual field loss to that incurred in the surface model
50
I r R r = ZpTpWc J r Psr (2.53)
If the a c t u a l f ield w ind ing c u r r e n t f rom e qu a t i on ( 2 . 5 1 ) is subst i tuted into the
e x p r e s s io n for f ield loss, the s u r fa ce e qu iv a le n t f ield winding re s is tan ce in
t e r m s of the a c t u a l va lue is.
Psr = Rr4( zr QrJ wr )^P^c
Using the expression for the winding resistance from Appendix ( 2 . 7 . 2 ) . and
making the sam e assumptions as above, the rotor surface resistance is given
by
TT^PCR kenÔRR
- 80 kpFR dR.
w h e r e the s u r f a c e r e s is t a n c e end fac to r . k e n d p R . is a ss u m ed to be equ a l to
that of the sta tor .
The DC field winding voltage, from equation ( 2 . 3 9 ) is
V r = 2p Wc Z R q R k ^ R /2 E r (2.55)
w h e r e the ro tor e i e c t r ic f ield s tr e n g th . E r . is as de f ined in sect ion 2 . 2 . 2 .
b) Salient pole
The salient pole rotor winding has one slot per pole and a winding factor that
Is approxim ately unity. The rotor surface current density in this case is.
/2 Z r I r
Jr ^ I
51
or in terms of the rotor conductor current density.
✓ 2 k p F R A s r K r
J r = I ( 2 . 5 6 )
As the geom etry is more com plica ted for the salient rotor construction, the
expression for the area of slot occupied by rotor conductors, (equation
2 . 2 3 ) is not easily simplified, and therefore the above equation for the rotor
surface current density is in its final form.
The equivalent surface res is tance of the salient pole rotor v/inding is obtained
as before by equating the actual and surface field loss. This gives.
"■pPsr = Rr
4ZR pWc
The expression for the winding res is tance , from Appendix ( 2 . 7 . 2 ) . is given
in terms of the m ean turn length, as
R r =PCR *mtR ZR p
kpFR A sr
The calculation of the m ean turn length. f ^ t R ' 'S based upon dimension "h"
of Fig. 2 . 9 . As the slot a rea is approximately triangular It was felt that a
reasonable estim ate of h would be given by one-th ird of the line GD.
T h u s ,
1h = - dRM sin 0p
and the mean turn length is th ere fo re .
4
fmtR = 2(Wc + wp + - dRM sin 0p )
52
This gives a surface resistance of
PCR Tp kendRR - 2 XPFR Ash
4dRMwhere kendRp = 1 + — - + sin 8p (2.58)Wc j Wq
The DC field vo ltage, for the salient rotor, from equation ( 2 . 5 5 ) is
Vr = 2p Wc Z r /2 E r (2.59)
53
2. 5 Induction m ach ine design method
A simplified flow chart illustrating the induction m achine design procedure is
shown in Fig. 2 . 1 1 .
Initially values have to be assigned to the independent variables, material and
core loss constants and packing factors. The maximum values of conductor
cu rren t density and co re flux density a re chosen empirically , according to
perm issib le losses and heating^^. An air gap length is also specified with
regard to the expected shock loadings to be imposed on the shaft, and the
environm ent in which the m ach ine will o pera te . The design method proceeds
initially by obtain ing a suitable stator core geom etry. This geometry Is
defined for a specified n um ber of poles in term s of the core outside d iam eter
do- slot depth d$. and the ratio of slot width to slot pitch, a . (section
2. 3. 1) .
For a known a ir gap length and a V shaped rotor slot as defined in section
2 . 3 . 4 . the m ach ine c ro ss -sec t ion is described completely, and hence the
total mass and length may be d eterm ined for any given core length.
A consideration is then m ade of the maximum surface current density and air
gap flux density a tta inab le by that geom etry , for the specified maximum
perm issib le values of core flux and conductor current densities.
The m ach ine p erfo rm ance at the design point for a specified shaft speed is
then d e term ined by evaluating the surface equivalent circuit. Fig. 2. 12. for
Increasing increm ents of slip. To illustrate the design method, the output
data for a four pole induction m ach ine is reproduced in F ig . 2. 13. This
design is rated at traction levels for a power output of 580 kW at 1500 rpm.
and m eets the requ irem ents of an Induction motor that is suitable for use in
54
a high speed d ie s e l-e le c tr ic locomotive application.
The ch o ice of flux and conductor current densities, together with the
geo m etr ica l constraints imposed on the design by the specification will be
described in m ore detail in the following chapter.
The results show that a lthough the effic iency d ecreases with increasing rotor
frequ en cy , the output power and power factor increase . The optimum
p erfo rm an ce occurs at the highest value of rotor frequency. This point is
defined by the maximum stator current loading.
The design techn iques discussed in this section enable the perform ance of a
m ach ine to be predicted for a given set of design variables at a set speed.
In o rder to investigate the motoring and braking characteristics of a design
over a range of operating frequenc ies , and with any suitable control
algorithm , a further program has been developed. Fig. 2. 14. This routine is
structured around the surface equivalent representation of the induction
m ach in e , and contains the option of including a braking resistor, for use in
the analysis of the traction motor braking perform ance described In Chapter
3.
Operating values of flux and conductor current density are not specified as an
Input to the p e rfo rm an ce program . This allows detail changes of m achine
geom etry to be m ad e to en ab le the design to be "trimmed".
55
Fig. 2.11 Induction Machine Design
A s i m p l i f i e d f l o w c h a r t ( t h e r e l e v a n t d e s i g n
e q u a t i o n s a r e sh o w n i n b r a c k e t s )
SET DESIGN PARAMETERS
max RMS stator conductor current density A/mm^
max RMS rotor conductor current density A/mm^
B max core flux density T
RPM synchronous speed rpm
P pole pairs
'"c core length m
stator core outside diameter m
stator slot depth m
g air gap length m
B slot width/slot pitch
7 coil pitch/pole pitch
*^PFS stator slot packing factor
kpFR rotor slot packing factor
^iS stator iron packing factor
*^iR rotor iron packing factor
k c ,X , 2 core loss constants
('cS stator conductor resistivity ohra-m
‘’cR rotor conductor resistivity ohm-m
stator conductor density kg/m^
^R rotor conductor density kg/m^
iron density
PRINT DESIGN PARAMETERS
kg/m^
1CALCULATE
pole pitch (2.19)
dc stator backing core depth (2.18)
dRM maximum rotor slot depth (2.33)
^ x c o '^ s c '^ps stator slot permeance correction,end ring
permeance correct ion,and pitch factors (A2. 7),(A2 .18),(A2.1)
mg-mR conductor mass (A 2 .5),(A 2 .17)
'"iS’'"iR iron mass (2.20),(2.34)
"’t o t total machine mass mg + m^ + m^g + m^j^
W t o t total machine length (core + end windings) '"c " (A2.3)
Pc core loss (2.21)
f supply frequency p*RPM/60
RMS air gap flux density (2.16)
ds maximum RMS stator surface current density (2.41)
dR maximum RMS rotor surface current density (2.47)Rendes-kend^g stator and rotor surface resistance (A2.6), (A2.9)kend^^.kend^^ and inductance end factors (A 2 .15),(A 2 .20)
57
surface equivalent circuit parameters (2.43),(2.44)
PRINT GEOMETRICAL, ELECTRICAL AND MAGNETIC DATA
ROTOR SURFACE CURRENT LOADING E X C E E D E D 'GOTO B
YES
NO
'STATOR SURFACE CURRENT LOADING EX CEEDED'GOTO B
YES
NO
CALCULATE MACHINE PERFORMANCEFROM THE SURFACE EQUIVALENT CIRCUIT (FIG.2.12)
RMS stator conductor current density (2.41)
RMS rotor conductor current density (2.47)
RMS air gap electric field strength (2.4)
RMS stator terminal electric field strength (2.7)air gap surface area 2pt w
input power ReINVA input VAIN
power factor
OUTtorque Pq .j,
LOSS
LOSS
PRINT PERFORMANCE DATA SPEED,f„,P, L O S S I N2 ' OUT
INCREMENT SLIP o
CALCULATE
J„ RMS STATOR SURFACE CURRENT DENSITY J„ + J,
CALCULATE f„ ROTOR FREQUENCY of
J_ RMS ROTOR SURFACE CURRENT DENSITY w t B /h z .
SPEED
J., RMS MAGNETISING SURFACE CURRENT DENSITY (2.3)
ROTOR SHAFT SPEED RPMx( 1 - <r )
58
GOTOA
YES MORESLIP
VALUES
NO
STOP
59
/ ss SS <SR
Eo E q
braking res isto r
/*S R<5
Fig 2.12 In d u c t io n Motor S u r f ac e Equiva len t C i rcu i t
(moto r ing and b ra k i ng )
Eo
B,R
(ss ■'s
Fig 2.16 Synchronous M o t o r Phasor Diagram ( f y = 9 0 ° )
60
Fig.2.13 Induction Machine Design, Output Data
62
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63
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64
Fig. 2.14 Induction Machine Performance, A simplified flow chart
SET MACHINE PARAMETERS
p pole pairs
^o stator core outside diameter m
^s stator slot depth m
"'c core length m
g air gap length m
B slot width/slot pitch
7 coil pitch/pole pitch
^PFS stator slot packing factor
^PFR rotor slot packing factor
^iS stator iron packing factor
^iR rotor iron packing factor
k^,x,z core loss constants
PcS stator conductor resistivity ohm-m
PcR rotor conductor resistivity ohm-m
stator conductor density kg/m^
«R rotor conductor density kg/m^
«i iron density
PRINT MACHINE PARAMETERS
kg/m^
CALCULATE
Tp pole pitch (2.19)
stator backing core depth (2.18)
dj j maximum rotor slot depth (2.33)
k ,k ,k stator slot permeance correction, end ringxco sc pspermeance correction and pitch factors (A 2 .7),(A 2 .18)
(A2.1)
mg , m^j, m^j,, m^Pj conductor and iron mass ( A2 . 5) , ( A2 .17 ), ( 2 . 20) , ( 2 . 34 )
m,j,Q,p total machine mass mg + m^ + m^^g + m^^^
total machine length (core + end windings) + ^oh
(A2.3)
core loss (2.21)
kendp^g , kendp^g stator and rotor surface resistance ( A2 . 6 ) , ( A2 .9 )
kendppj^ , kendj^P^ and inductance end factors ( A2 .15 ) , ( A2 . 20)
pgg.jfgg I (2.43),(2.44)
‘ SR’’ SR I surface equivalent circuit parameters ( 2 .49 ) , ( 2 .50)
L I (2.6)
PRINT Tp*‘ c’^ R M ’"’s ’'"R’'"iS''"iR’"’TOT''^TOT
66
CALCULATE
slip
supply angular frequency rads/sec
rotor speedspeed
EVALUATE EQUIVALENT CIRCUIT TO FIND
stator surface current density A/m
rotor surface current density A/m
air gap electric field strength V/m
NO
Jp BRAKING RESISTOR SURFACE CURRENT DENSITY
YES
CALCULATE PERFORMANCE CHARACTERISTICS AT CONTROL POINT
maximum core flux density (2.4),(2.16)
stator conductor current density (2.41)
rotor conductor current density (2.47)
output power pOUTtorque P p/( l-o)wtotal losses P,LOSSinput power ReIN
SB
BRAKING RESISTOR INCLUDED
IB = 1?
J SUPPLY SURFACE CURRENT DENSITY
IB
SB
brake resistor option
brake resistor excluded
braking resistance in surface terms
stator terminal electric field strength
supply frequency
brake resistor included
rotor frequency
SET CONTROL POINT PARAMETERS (MOTORING OR BRAKING)
V/m
ohm
HzHz
67
supply VASUPP.VAbraking resistor loss pg^
angle between E and J„
OUT ^ OUT LOSS
Pg,T,pf.n.SUPPy^PRINT P IN’ OUT* LOS:
'ERFORMANCE DATA
YESGOTO
NO
STOP
68
2, 6 Synchronous m ach ine design method
The synchronous m ach ine design method illustrated by the flow chart of Fig.
2. 15. p ro ceed s initially in a s im ilar m an n er to the induction m achine. The
m ethod requ ires the sam e input variables to be specified, with the addition of
one extra variab le for the salient pole rotor construction. This Is the ratio of
pole a rc to pole p itch, a , and is required for the calculation of the d and q
axis m agnetis ing Inductance factors C(j and Cq.
Throughout the design process the synchronous m achines of both the round
rotor and sa lient pole type are assum ed to be controlled at a torque angle ,
6j , of 90 d e g re e s . This will produce the maximum excitation torque per unit
cu rren t. Fig. 2. 16.
The stator geom etry Is determ ined as before, and again sets the maximum
perm issab le values of air gap flux density, Bg, and stator surface current
density, J^, for a specified core flux and conductor current density. To find
a starting point In the Iterative design process, the a ir gap length is Increased
in steps until the stator com ponent of air gap flux, 8$ , is equal to or less
than the m axim um permitted air gap flux density, Bg.
From equation ( 2 . 10) for the round rotor m ach in e ,
and for the salient pole m ach in e , using equation (2 . 12)
Mo'^p^q'^s
~ ng
69
As the torque an g le Is held constant at 90 d e g re e s , the stator and rotor
com ponents of a i r -g a p flux are at right angles and therefore the rotor
com ponent of the air gap flux can be found from.
B r =
Having de term ined an air gap length that satisfies the above condition, it is
possible to define the rotor c ro ss -sec t ion (sections 2 . 3 . 2 and 2 . 3 . 4 ) and
hence ca lcu la te the availab le slot a re a . The value of rotor surface current
density. J p . requ ired to set up the rotor com ponent of a ir -g a p flux. Bp, is
then d e term ined using equations ( 2 . 1 0 b ) and ( 2 . 1 4 ) for the round and
salient pole rotor structures respectively. Thus, for the round rotor
m achine
ng B r
" Wo Tp
and for the salient pole m a c h in e ,
ng Br
Mo " p
The rotor slot a re a availab le must be large enough to accom m odate a total
conductor c ro s s -s e c t io n , that is able to support the required level of rotor
surface current load ing , J p , at the specified conductor current density, Kp.
If sufficient slot a re a is ava ilab le , the depth to which it need be filled with
conductors, for operation at the given rotor conductor current density is
found ( d p ) . if both the a i r -g a p and rotor slot a rea criteria are met, the
m ach in e p er fo rm an ce for that part icular a i r -g a p length is evaluated.
70
The a i r -g a p length is then increased in increm ents of 1 m m. for each
subsequent pass around the design loop. For each Increase in the a ir -g a p
length, the stator com ponent of flux loading. Bg. will be reduced and the
rotor flux load ing . Bp . will increase for a fixed value of a ir gap flux density.
B g .
For a fixed stator c ro ss -sec tion and core length, and a maximum value of
stator surface cu rren t loading, the output power will increase as the rotor
surface cu rren t and flux loadings in crease with progressive increases in
a ir -g a p length. The output power will continue to rise until a gap length is
reached for which there is insufficient rotor slot space available to support
the required value of rotor surface current loading, for a fixed rotor
conductor curren t density. The power factor also increases with increasing
gap length. This is due to a consequent decrease in the surface
m agnetis ing indu ctance . The best design was therefore taken at the
maximum allowable gap.
To illustrate the design m ethod, the output data for both a round rotor and
salient pole design is reproduced in Fig. 2. 17. These designs again
represent a m ach in e that Is suitable for the traction application described in
the following ch ap ter .
As for the induction m ach in e , there Is a need to predict the perform ance of
the synchronous m ach in e for any control a lgorithm , under both motoring and
braking conditions. The flow chart of Fig. 2. 18 illustrates a method which
enables this to be done. The method is based on surface equivalent
quantities, and provision has been m ade for the inclusion of a resistor for
analysis of the braking duty. All input variables regarding the m achine
geom etry and m ateria l constants a re unchanged from those used in the
71
design p rocess , with the exception of the a i r -g a p length which is now
defined initially. The rotor slot depth may also be defined, or the maximum
slot depth used, depending upon the setting of the flag IS. No stipulation Is
m ad e reg ard ing the choice of stator conductor current density or maximum
core flux density. These quantities a re evaluated as a consequence of the
control point param eters . This approach again enables the effect of small
g eo m etr ica l ch an g es to be evaluated so that the design can be "trimmed".
72
F i g . 2 . 1 5 S y n c h r o n o u s M a c h i n e D e s i g n
A s i m p l i f i e d f l o w c h a r t
* r o u n d r o t o r
* * s a l i e n t p o l e
SET DESIGN PARAMETERS
Kgmax RMS stator conductor current density A/mm^
ymax RMS rotor conductor current density A/mm^
B max core flux density T
RPM synchronous speed rpm
P pole pairs
"'c core length m
stator core outside diameter mstator slot depth m
6 slot width/slot pitch
1 coil pitch/pole pitch
a pole arc/pole pitch
kpFS stator slot packing factor
kpFR rotor slot packing factor
^iS stator iron packing factor
^iR rotor iron packing factor
^c,x Z core loss constants
^CS stator conductor resistivity ohm-m
• CR rotor conductor resistivity ohm-m
' sstator conductor density kg/m^
^R rotor conductor density kg/m^
«1iron density
PRINT DESIGN PARAMETERS
kg/m^
1 .................CALCULATE
pole pitch (2.19)
stator backing core depth (2.18)
'^ppole body width 2d^**
*^xco ' ps slot permeance correction and pitch factors (A2.7),(A2.i)
C d ' C q magnetising inductance d and q axis factors (A2.30).(A2.31)**
"’S ’'"iS stator conductor and iron mass (A 2 .5),(2.20)
'^ T O T total machine length (core + end windings) w c + 2 ' o h ( A 2 . 3 )
Pc core loss (2.21)
lu
supply frequency pRPM/60
RMS air gap flux density (2.16)maximum RMS stator surface current density (2.41)
kendj^g, kendj^g stator surface resistance and inductance endfactors (A 2 .6),(A 2 .9)
stator surface resistance and inductance (2.43),(2.44)
PRINT GEOMETRICAL, ELECTRICAL AND MAGNETIC DATA
YESGOTO A'AIR GAP "TOO
SMALL'NO
NOGOTO B
'SLOT AREA REQUIRED TOO L A R G E '
YES
MACHINE PERFORMANCE
PHASOR DIAGRAM REPRESENTATION (6. = 90 )
CALCULATE
rotor slot depth required
rotor conductor mass (A2.25)*,(A2.22)**
iRtotal machine mass mg + m^ + m^g
slot area, rotor surface resistance factor (2.23, A2.23)iRT O T
AgR'kendRR kendj R round rotor onlykend RS
INCREMENT AIR GAP LENTH CALCULATE STATOR COMPONENT OF AIR GAP FLUX. Eg(2.10a)* (2.13b)**
IS THE AVAILABLE SLOT AREA LARGE ENOUGH TO SUPPORT THE REQUIRED ROTOR SURFACE CURRENT DENSITY, Jo, AT TgE SPECIFIED ROTOR CONDUCTOR CURRENT DENSITY, Ko
RM
rotor component of air gap flux
maximum rotor slot depth (2.28)*, (2.22)**
rotor radius (2.22a)
rotor surface current density (2.10b)*,(2.14)
CALCULATE
75
air gap surface area
I R Q S ^ S Sstator terminal electric field strength
SS S
rotor surface resistance (2.54)*,(2.57)**SRrotor terminal electric field strength PqoJ,
stator and rotor conductor losses
LOSSoutput power (2.11)*,(2.15)**OUTtorque P p/2JTf
efficiency + ^ l OSSpower factor E /E
PRINT PERFORMANCE DATA g ,d, LOSS OUT 'tot ' ""a
YES MOREGAPS
GOTO A
NO
STOP
76
Fig.2.17a Synchronous Machine Design, Output Data (Round Rotor)
7 8
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79
Fig. 2.17b Synchronous Machine Design, Output Data (Salient Pole)
81
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82
F i g . 2 . 1 8 S y n c h r o n o u s M a c h i n e P e r f o r m a n c e
A s i m p l i f i e d f l o w c h a r t
p
s e t MACHINE PARAMETERS
pole pairs
d stator core outside diameter mo
^s stator slot depth m
' c core length m
K air gap length m
B slot width/slot pitch
7 coil pitch/pole pitch
a pole arc/pole pitch
^PFS stator slot packing factor
^PFR rotor slot packing factor
*^iS stator iron packing factor
^iR rotor iron packing factor
K . core loss constantsc ■ x , z
‘ CS stator conductor resistivity ohm-m
^CR rotor conductor resistivity ohm-m
«S stator conductor density kg/m^
«R rotor conductor density kg/m^
«i iron density kg/m^
^S 1 - for given rotor slot depth
0 - to use maximum slot depth
PRINT MACHINE PARAMETERS
IS MAX ROTOR SLOT DEPT
TO BE USED IS=0?
YES
SET SLOTDEPTH ^R
^xco’^ps
o
RM
TOT
CALCULATE
pole pitch (2.19)
stator backing core depth (2.18)
pole body width 2d^**
slot permeance correction and Ditch factors (A 2 .7).(A 2 .11
magnetic inductance d and q axis factors (A 2 .30),(A 2 .31)**
rotor radius (2.22a)
max slot depth (2.28)*,(2.22)**
total machine length (core + end windings) (A2.3)
84
IS GIVEN ROTOR SLOT DEPTH TO BE
USED 18=1?
YES
A
SR kend„q,kend'RS 'kendRR
^SS’^SS
*SR^m
"’i S ’"’iR
'"t o t
LS
CALCULATE SURFACE RESISTANCES, INDUCTANCES AND MASS
air gap surface area 2pTpW^
rotor slot area (2.23)**
surface resistance and (A 2 .6),(A 2 .9)
inductance end factors (A 2 .26)*,(A 2 .23)**
surface resistances and (2.43),(2.44)
inductances (2.54)*,(2.57)**,(2.6)
conductor mass (A 2 .5).(A 2 .25)*,(A 2 .221 * *
iron mass (2.20),(2.311*,r2.26)**
total machine mass
PRINT Tp, d^ , Wp** , d ^ ^ , dj^, mg ,mj^. m ^ g , m^
^a
SET CONTROL POINT PARAMETERS (MOTORING OR BRAKING)
stator terminal electric field strength V/m
f supply frequency Hz
6t torque angle deg
^R rotor conductor current density A/mm^
IB brake resistor option
"SB
1 - braking resistor included
0 - braking resistor excluded
braking resistance in surface terms ohm
Speed
Zs
"R
' RC
CALCULATE
supply angular frequency
synchronous speed
complex magnetising impedance
Stator impedance (Pgg + jw iqq)
rads/sec
60 f /p rpm
SS'rotor surface current density (2.52)*,(2.56)**
complex value cos a - j sin 6^ (Fig.2.6)
85
electric field strength due to rotor current
RC
YES
NO
CALCULATE PERFORMANCE CHARACTERISTICS AT CONTROL POINT
power factor angleac
Fig.(2.16)Jgg + Jg cos $ + j Jg sin * output power (2.11)*,(2.16)**
ac
OUTtorque P p/rotor terminal electric field strength
air gap electric field strength Egc s scRMS air gap flux density (2.4)
maximum core flux density (2.16)core loss (2.21)
stator loss
rotor loss
brake resistor loss pg^ Jg A
stator conductor current density (2.41)
power factor cos ((>
INsupply VAVA IN ac actotal lossesLOSSefficiency ^OUT^^ OUT LOSS
PRINT PERFORMANCE DATA S p e e d .f ,6„,E E P LOSSIN' OUTbraking only)
YESMORE CASES
GOTO A
NO
3 R A K I N G \ RESISTOR INCLUDED V IB=1? y
ITERATE TO FIND APPROPRIATE SURFACE CURRENT DENSITY, J^, FOR GIVEN STATOR TERMINAL ELECTRIC FIELD STRENGTH, E
STOP
86
2. 7 APPEN D IC ES
2. 7. 1 List ot sym bols used in the m a c h in e d e s ig n p ro ce ss
F in a l sub scrip t S or R d e n o te s s ta to r or ro to r quantity .
C O N S T A N T S and F A C T O R S
p number of pole pairs
winding factor kp coil pitch factor
^xt'^xco slot permeance correction factorsksc squirre1-cage end ring permeance correction factor
kgndR surface resistance end factorkgndL surface inductance end factorkpF slot packing factorki iron packing factork g tooth flux fringing factorkv vent flux fringing factorkvL vent flux fringing factor for slot leakagenv number of ventsKc,x,z constants for core loss calculation6 conductor density
6j_ iron density6er squirrel cage end ring densitykq distribution factorPq magnetic constant 4tt x 1 0 “ H/m
88
DIMENSIONS
T Slot pitchTp pole pitch
Tc coil pitch
Ta pole axeg air gap length
d slot depthw slot widthwt tooth widthw% slot opening (semi-closed slots)Wp pole width ( salient rotor)
fioh slot conductor overhangAg slot axeaw^ cooling vent widthTy vent pitchdq. stator core depthdo stator core outside diameter0 angle between core and end winding conductor for
diamond-ended coils
□er mean diameter of squirrel cage end ringA^r cross-sectional area of end ring
ter thickness of end ringWer width of end ringg^t mean length of coil turnWq core length
0p half pole angle (salient rotor )m conductor massmj iron massA air gap surface area
89
ELECTRICAL AND MAGNETIC
AB maximum core flux density TB g RMS air-gap flux density T
Pc conductor resistivity DM
Ps conductor surface resistance n
*s surface leakage inductance H
*in surface magnetising inductance HR phase resistance n
L phase lesücage inductance H
Lm phase magnetising inductance H
P slot width/slot pitch
y coil pitch/pole pitcha pole arc/pole pitchz series conductors per slot
q slots per pole phaseA specific permeanceK conductor current density A/mm^J surface current density A/MV terminal voltage V
E electric field strength V/MP conductor loss W
Pc core loss W
PqUT output power WT torque Nm5'p torque angle degreew supply angular frequency rad/secW r angular frequency of rotor currents rad/seca slip puRe indicates the complex real part
90
* indicates the complex conjugate
e instantaneous electric field strength V/mb instantaneous magnetic flux density T
j instantaneous surface current density A/m
M magneto-motive force AT
I RMS phase current AQ total number of slotsf supply frequency Hz
f2 rotor frequency Hz
91
2 . 1 . 2 Machine parameters
F o rm u la e for m a c h in e w ind ing r e s is ta n c e s . In d u c ta n c e s and m asses a re
d er ived In this s e c t io n . M ost of th e e x p re s s io n s a re b ased on those given In
r e fe r e n c e s 3 3 . 3 4 . Fo llow ing the d er iva t io n of e a c h fo rm u la , a s im plif ied
fo rm is g iven w h ich is m o re s u ita b le for In itia l d es ig n use.
1 ) M a g n e t is in g In d u c ta n c e
T h e m a g n e t is in g in d u c ta n c e s e e n by o n e p h a s e of a b a la n c e d t h r e e -p h a s e
w ind ing driv ing flux a c ro s s a u n ifo rm gap.l^j(A2. la). Is:
P-o6( Zs^S^WS )^TpWQP
TT^kvkgg
F rin g in g of the tooth flux is a c c o u n te d for by a fac to r kg. For o n e slotted
m e m b e r :
T s(5g + wg)
Tg(5g + Wg) - Wgk g = --------------- (Open slots)
Tg( 4. 4g 4- 0.75W 2 )k g = ---------------------- (Semi-closed slots)
T g ( 4 .4 g + 0 . 7 5 w j ) - wf
If both the s tator a nd rotor c o re a r e s lo tted then kg Is eva luated for both,
and the ir re s u lta n t is g iven by
^g “ ^gS ^ ^gR
A fac to r ky takes a c c o u n t of flux f r in g in g at ra d ia l ven ts . Fig. A 2 . lb . For a
ven ted m e m b e r
T v (5g + Wv) k y - -----------------------------
Ty( 5g 4- Wy ) - Wy
92
where the vent pitch is;
Wf, + Wx
It both the stator and rotor contain cooling vents, then ky is evaluated for
both and the resultant is given by:
Jcy - kys ^
A simplified form of the expression for can be obtained by assuming that
ky = kg = 1. Also, s ince the distribution factor will be close to unity, the
winding factor can be rep laced by the pitch factor:
7rrkps = s in — ( A 2 .1 )
where 7 is the ratio of coil pitch to pole pitch
T h e f inal form is th e n .
6( Zgqgkpg )^WqP Mo'^pLm = (A2.2)
Td grr^
2) S ta to r w in d in g re s is ta n c e and m ass
The p h a s e r e s is ta n c e and total m ass of the th re e ph ase stator w inding Is
given by the exp ress io n s:
Pcs mts^s^sP
kpFsAss
= 3*mts^sqsP^PFsAsS
93
Each staler turn has a m ean length, the d im ensions of which are defined in
Fig. A2. 2a , of
2 (oh( jn t s " 2 (W c + 2 f io h s 0 ^ 1 . 6 rrdg )
These form ulae can be usefully simplified by Ignoring the detail of the nose
and slot conductor overhang. If a m in im um en d -tu rn configuration Is
assum ed. F ig . A 2 .2 b . then
Wssin 9 - P - —
PTc 1ajid * - — ( A2.3 )
2 V l - P^
If the slot Is assum ed to be rec tang u la r , then the slot area Agg = Wgdg.
and the final expression for the phase res is tance is.
6Pcs^s9s^c^endR 5
kpFsdsP^p
and the conductor m ass.
(A2.4)
iSspkpFs^Tp^^s'^c^endRS ( A2.5 )
2 (ohwhere kendRS = 1 (A2.6)
3) Stator winding leakage inductance
Stator winding phase leakage Is given by
23 Mo^s2Pqs ( ^3 ^e )
94
The factor k\/L accounts for the fringing of leakage flux due to the presence
of radial vents and is given by:
T y ( 5 W i + 2 W y )
kvL =T y ( 5 W 2 + 2 W y ) - 2W y
V n-17 + L
E n d-w ind ing specific p e rm e a n c e can be estim ated from (F ig . A 2 .2 a )
Wf
The slot specific p e rm e a n c e is given by
~ ^ x t ^ s t ^xco^sco
where
Agt = specific permeance of the slot cüDove the conductor^sco “ specific permeance of the slot containing conductor
^3 d l
Fig.A2.3
For an open slot Agt = — and AWo ' SCO 3w
d3 2d 4 dgand for a semi-closed slot Aot- = — + + —Wg Wi + Wg wi
d land A,SCO 3w<
The factors kyt and kyco co rrect for short-p itch ing of the coils and are
defined by:
95
2 Tq 1 Tq 2— < — < I and for — < — < —3 T p 2 T p 3
1 ■3Tc 1 9T ckxt = 4 ---+ 1iTp kxt = 4 + 1
[ T p( A 2 . 7 )
‘•xco1
169 T ,
+ t ^XCO1
1618Tc 7
+ %Tr
The following assumptions are m ade to obtain a simplified expression for the
phase leakage Inductance
I) k\/L = 1
il) the slot conductor overhang ( fighs^ is ignored
ill) the winding has a m inimum end turn configuration so that
p r Q w*(oh where p =
2 A - p^
iv) the slot is rec tangu lar and com plete ly filled
i.e. A,3w<
- )cxco
v) the distribution factor is close to unity
This gives (us ing SpsTg = Tp)
B( Z s 9 s ) ^cP Mo^xco^s
3/3 kendLS
where kgndLS ^ 1 +0 . GkpglohTp#
^ckxco^s
( A2 . 8 )
(A2.9)
96
4) Squirrel cage phase res istance and mass
Fig. A 2 . 4 shows the relationship between squirrel cage en d -r In g and bar
currents. From the geom etry of the d iagram
e ( A2 .10 )se
2 s in
where ©ge the slot ang le in e lectr ica l d eg rees , i .e .
2n©<
6qR
The resistance of an individual rotor bar is:
PcR(Wc + 2 lohR )
For that part of the end ring between ad jacent slots (F ig . A 2 .5 )
Per^Der (A2.ll)® AerGPqR
The resistance of the winding can now be found by equating losses i. e.
IRRR = 2P9R(iR^b + 2 I e r e )
Using equations (A 2 . 10) . ( A 2 .1 1 ) and (A 2 . 12) and assuming that ©ge is
small gives the result:
R r - 2pqRPc r C'^c 2 lo h R ) P e r^ e r ^ q R
+kpFR Asr TrAerP
(A 2.13 )
The winding has a total of 6 qpp rotor bars and 12 qpp en d -r in g bars
The total conductor mass in the rotor is therefore:
mR = 6pqRkpFRAsR(Wc + 2 8o> r)6r + 2rrDerAer®er (A2.14)97
Simpler forms of equat ions ( A 2 . 1 3 ) and (A2. 14) can be obtained by making
the following assumptions:
I) (o h R Is neglig ib le
II) bars and end rings are m ade of the sam e material
iii) the peak current density in the e n d -r in g s is equal to that in the
bars, i . e .
iv) the num ber of rotor slots is equal to the number of stator slots
v) the length of an e n d -r in g segm ent is equal to the slot pitch, i .e .
Der^
6qsP
With these sim plif ications, equations (A 2 . 13) and (A2. 14) become
SqgWcP PcR^sRr = kpFR^SR kendRR
mp = 26RpkpFR ” AsRW^kendRR
2 T r
W h e re k e n d R R ^ 1 +TTWr
(A2.15)
Consideration of the rotor slot geom etry (2 . 3 .4 ) shows that, for a tapered
slot with paralle l tee th , the slot area
1 -
ZdRM
With an effective slot depth d , (equation 2 3 0 ) . defined as
98
% = dR 1 -dR
2dRM
then the above equation becom es
R r = Tp kpFRdR/3 ^endRR (A2.16)
BttR = 2 6 RpkpRR/3TpdRWckg ndRR ( A2 .17 )
5) Squirrel cage phase leakage inductance
By considering the bar and end ring curren ts , as in section 4, and also by
equating VAr, the phase leakage inductance is found to be
L r = 2pq^^o (As + Ae )
In this expression k\/L can be found by the methods discussed in section 3.
Form ulae for slot specific p e rm ean ce (Ag) are also given in section 3 but In
this case the correction factors k^co acd k^t a re set to unity.
The e n d -r in g specific p e rm e a n c e is given by:
Wf
^sc^er^1.32 lohR + 2p
where kg^ ^ 0.36 for p -= 1and kgc ^ 0.18 for p > 1 .
(A2.18)
A simplified expression for the slot leakage inductance can be obtained
with the following assumptions:
99
i) k\/L = 1
il) qp = qs
ill) Ignoring the slot conductor overhang ( *o h R )
Iv) the length of each e n d -r in g segm ent is equal to the slot pitch
i . e .
s -6q[sP
It is a lso assum ed that for the purposes of slot leakage, the tapered slot
used can be considered to be rec tan g u la r^^ , i. e.
dR- —3Wg
With these s im plif ications, the slot leakage inductance is given by the
expression
Lr = %endLR (A2.19)
where kendLR" + ~ d ^ ^ ( A2.20 )
6) Salient pole field winding res is tance and mass
Th e res is tance of a salient pole winding is
R r =
PcR^mtRZRP
kpFR Asr
and its mass is
^ (mtRGRPkpFRAsR
100
B ecause of the com plex slot geom etry it is not possible to express A gp in
terms of effective widths and depths. The only simplification that can be
m ade in this c a s e Is to use the form ula for turn length ( 2 . 5 8 ) from
(sec tio n 2 . 4 . 4 ) . In this case
22ZrPWc PcRTp
Rr = — kpFRAsR
thr = 26RpXppRAsRWc)CendRR (A2.22)
4dRMWhere k^ndRR = 1 + " + sin ep (A2.23)
W p T7
2 Sin 0p where ©p 2pT
and dRM = Ro - 7 — where ©p
7) R o u n d -ro to r field winding res is tance and mass
The form ulae for the round rotor case can be obtained from those of the
salient case by considering z r slot conductors to be distributed into qp slots.
This gives.
PcRlmtRSRZRPR r =
kpFR As r
= %tRGRqRPkppRAsR
101
Simpli fied forms are obtained, by making the following assumptions
i) co i ls have d ia m o n d e n d s so that e q u a tio n A 2 . 3 is a p p ro p r ia te .
T h e n o se s and c o n d u c to r o v e rh a n g s a re ignored and a m in im um
e n d - t u r n c o n f ig u ra t io n is a s s u m e d ,
il) qR slots span o n e p o le .
ili) T R = T g .
iv) slot a r e a A g R = Wgd r ^
This g ives:
X2(3qgZR) PWc PcR
Rr = - kpFRdRg kendRR (A2.24)
TtiR = 26ppkppR/3TpdR.Wc)CendRR (A2.25)
2*ohwhere kendRR ^ 1 + (A2.26)
8 ) R e fe r re d q u an t it ies
F or the in v e r te r fed m a c h in e p e r fo r m a n c e p red ic t io n s of C h ap ter 4 . the
r e fe r re d v a lu e s of the rotor q u a n t it ie s a re re q u ire d .
F or the s q u ir re l c a g e Ind u ctio n m a c h in e , the turns ratio is given by
n = ( A 2 . 2 7 )qR
F o r the round ro tor s y n c h ro n o u s m a c h in e :
V3 zs qs kW Sn - ----------------r — ( A 2 . 2 8 )V2 Zr qR kwR
102
and for the sal ient pole case:
V3 Zs qs kws
103
stator core
lamina tions
open
semi-c losed slot
rotor core
laminations
Fig A 2.1 D imens ions for slot and ve n t f r i ng ing fac to rs
nose lengtti
1 6TTdc
oti
Ts
Q basic coil b coil arrangement for
minimum end turn
Fig A 2.2 Dimensions fo r the c a l c u l a t i o n of mean tu rn l e n g th
d
d
M
open slot sem i-open slot
Fig A 2.3 S lo t dimensions f o r the ca l cu la t i on of speci f i c permeance
104
Ie2
'se
Fig A 2 .4 Squi r re l Cage ro t o r winding cu r r en ts
'ohR ter
Fig A 2.5 Dimensions of Squi r re l Cage end r ing
105
2 . 7 . 3 Calcu lation of the salient pole factors O p and C q
T h e s a l ie n t ro to r g ives r ise to non s in u s o id a l a i r - g a p flux c o m p o n e n ts .
H o w e v e r the s in u s o id a l ly d is tr ibu ted w in d in g c o n d u c to rs will respond only to
flux d e n s ity c o m p o n e n ts hav ing the s a m e pole p itch . This m e an s that only
the fu n d a m e n ta l c o m p o n e n ts n e e d be c o n s id e r e d and these a re a c c o u n te d for
by the fa c to rs 0 ^ a nd C q .
T h e fie ld w in d in g p ro d u c e s a flux d en s ity d is tr ibu t ion as shown in F ig . A 2 . 6.
T h e a m p l i tu d e of the fu n d a m e n ta l c o m p o n e n t is given by
1~ Bfj — ( orrr + sin orrr )
w h e re B r is the a m p li tu d e p ro d u c e d In a un iform g ap of length , g . and a Is
th e ra tio of p o le a rc to po le p itch . Th is exp re ss io n can be written as
Br = Br Cd
where C(3 = — ( a n + sin a n ) (A2.30)
T h e re s u lta n t a r m a tu r e flux is c o n s id e r e d to be due to two s e p a r a te
c o m p o n e n ts a lo n g the two axes of s y m m e try of the rotor. T h e a m p litu d e of
the c o m p o n e n t w h ich Is c e n t re d on the p o le s . F ig . A 2 . 6 b , Is
B(3 = B g C d
a n d at the c e n t r e of the In te rp o la r s p a c e . F ig . A 2 . 6c . the c o r re s p o n d in g
c o m p o n e n t is
B q Bg C q
1where Cq ^ - ( a n - sin a n ) (A2 31)
106
and Bg is the am plitude of the a i r -g a p flux density that would be produced by
the stator winding In a uniform gap of length , g.
107
( q)
( b )
la) pole geom etry
I b) d - a x is f lux d i s t r ib u t io n
(cl q - ax is flux d is t r ib u t io n
Fig A2.6 Po le geometry for the ca l cu la t ion of the factors Cd and Cq
108
C H A PTER 3
AC TRA CTIO N M O TO R DESIGN A N D PERFORM ANCE PREDICTIONS
FOR A H IGH S P E E D DIESEL LOCOMOTIVE
3. 1 Introduction
Traction motors used by British Rail range in output from less than lOOKW
on s o m e multiple units, up to almost IM W on main line locomotives.
Without exception they a re DC m ach ines , seperate ly excited, series or
com p ou nd wound. T h ese motors a re operated at a constant torque from
standstill up to a k n e e -p o in t speed , thereafter the torque falls rapidly with
Increas in g speed as the air gap flux density decreases .
The s m a lle r DC m otors a re usually axle hung via single reduction gearing.
Larger and h eav ie r motors can n o t be axle hung as the unsprung mass
would be excessive. T h e la rger motors are therefore bogie mounted and
drive the axle through gearing and a flexible coupling. For high speed44
applications such as the Advanced P assenger Train . the drive to each
powered axle is through a m echan ica l transmission from the traction
m otors, which a re mounted in the power c a r body. Fig 3 . 1 . Power is
transm itted via a body mounted transfer gearbox, cardan shaft, final drive
gearbox , which is fully suspended on the bogie fram e, and a flexible quill
to the axle.
R e p la c e m e n t of the DC motor with an AC induction or synchronous
m ach in e has m any advantages.
R educed weight - T h e removal of the com m utator saves weight directly
and also allows the motor to run at h igher speeds to give a sm aller
m ach in e for a given power output.
110
NrNr<Xj
Oo
op
111
Less m aln ta lnancG - As an AC m ach ine has no com m utator or brushes
to Insp ect, the resulting design Is genera lly s im pler and more rugged.
This leads to g re a te r reliability, less m aln ta ln an ce and a lower first cost.
A syn chronous m ach in e will of course require brushes to feed the rotor
slip r ings , but this Is a much s im pler a rran g em en t.
Increase d pow er at high speeds - C om m utator motors are frequently
unable to supply rated power up to maximum speed because the reactance
voltages b e c o m e too large.
U nfortunate ly , the basic torque - speed character is tic for a single cage
Induction m otor driven from a constant voltage, constant frequency
supply. Is c lear ly unsuitable for traction applications. Useful torque Is only
ava ilab le over a very restricted speed range n ear synchronous speed, and
the starting torque and effic iency are poor. By variation of the supply
voltage and frequ en cy however, a family of torque curves can be produced
to give a com posite curve of the required form . Fig 3 . 2 .
The princ ip le d raw back of AC traction systems Is the need for a variable
voltage, var iab le freq uency supply. Over the last d ecad e or so, advances
In pow er sem ico n d u cto r technology have m ade Inverters with suitable
power ratings a practical reality. Various basic Inverter circuit
configurations a re em ployed, each offering particular advantages. This
aspect of the traction system will be looked at In g reater depth In the
following ch ap te r .
17, 18The Brush Hawk was the first attempt In Britain to evaluate an AC
traction system under practical conditions using thyristor Inverters, and
the fa ilure of this project can be attributed to the lack of suitable
112
sem ico n d u cto r devices being ava ilab le at that tim e. As better devices have
ap p eared re s e a rc h Into the prospects of three phase traction systems has
In crease d . This research has produced m any experimental railway vehicles 19 20
for evaluation . som e of which have now reached production status
It has b een proposed that the m ech an ica l transmission system and DC
traction m otor be rep laced by a bogle mounted AC m ach ine, to produce
a variab le speed drive for a high speed d ie s e l-e lec tr ic locomotive. A
highly rated AC m ach in e would seem well suited to this particular
application w h ere the available sp ace Is restricted and where the weight
must be kept to a m in im um .
In this c h a p te r designs are p resented for squirrel cage Induction and slip
ring synchronous m ach ines that will m eet the motoring and braking
requ irem ents of this application.
The relative sizes of the m ach ine and power supply are dependant upon
the m an n er In which the system Is contro lled . It was therefore decided to
consider the designs resulting from two com m only used control schem es .
In o rder to Illustrate the resulting motor weights and power supply
capacities.
The techniques d escribed In the previous chapter a re used to dem onstrate
the Influence of the pole num ber on the main requirem ent for a minimum
size and w eight design . From this com parison , which Includes an
assessm ent of the power factor and losses , six competing designs (o n e
for each m a c h in e type and control m ethod) , a re chosen for further study
at what Is co n s id ered to be the optim um pole num ber.
113
In section 3 . 3 p erfo rm ance predictions a re shown for these six designs
over the motoring and braking regions of the traction character is tic , and
an appra isa l Is m ad e of their suitability for this traction application.
The m ach ines presented In this ch ap ter have been designed assuming
sinusoidal supply conditions. As In p ractice the traction motors would be
supplied by an Inverter of an appropria te type, the most suitable Induction
and synchronous designs are carr ied forward Into C hapter 4 . where their
perfo rm ance Is p red icted when they a re fed with a voltage or current
source Inverter.
1U
3. 2 The traction motor character is tic and control strategy
The traction m otor specification for a high speed d ie s e l-e lec tr ic locomotive
is as follows:
1 The train configuration is to consist of one power and seven trailing
cars giving a total mass of 320 tonne.
2 T h e duty cycle for a typical journey from London. Euston to Glasgow
Centra l is to be as shown in Appendix 3 . 5 . 1 .
3 Pow er c a r to have four traction motors.
4 A constant force of 36 KN required from standstill to 55 K m /h r .
5 A constant power of 2. 2 fVIW required from 55 to 225 K m /h r .
6 A constant braking force of 22. 5 KN required from 225 K m /h r to a
speed of less than 55 K m /h r .
7 M axim um m otor d imensions to be 900 mm in length and 600 mm in
d iam ete r .
8 M otor weight to be less than 1 .6 tonne.
9 M axim um m otor speed to be about 6000 rpm.
10 Maxim um AC line voltages to be about 1 KV.
11 Maxim um DC field voltages to be about 1 lOV.
12 C urren t densities to be suitable for class H insulation.
13 W heel d iam eters a re 853 mm ( n e w ) , and 823 mm (w o rn ) .
It is des irab le to use as high a motor speed as is possible. This implies
the use of an axle mounted gearbox. On the axle, the pinion d iam eter is
limited by the w hee ls ize . and thus the highest speed ratio available is
determ ined by the m inimum practical d iam eter of the motor pinion. The
main considerations for determ ining the minimum pinion diam eter are the
m ateria l s trength, and the num ber of teeth meshing with the axle pinion.
115
36.A ratio of approxim ately 4: 1 Is considered to be the practical limit
In o rd e r to give a round figure of 1500 rpm at 55 K m /h r with a new
w h ee l, a g e a r ratio of 4 . 3 9 : 1 was adopted. The motor shaft speeds for
the upper and lower limits on wheelsize a re given In table 3 .1 .
Train Wheel Motorspeed ( K m /h r ) d iam eter (m m ) speed (rpm )
225 853 6143823 6367
55 853 1500823 1556
Table 3. 1 Motor shaft speed with a 4. 39: 1 gearbox
It can be seen that the maximum motor speed is 6367 rpm at a train
speed of 225 K m /h r . This value Is felt to be close enough to the specified
m axim um m otor speed of about 6000 rpm.
The traction m otor character is tics for one motor are shown in Fig 3. 2. In
the motoring m ode a constant torque of 3. 7 KNm Is required to produce
the desired constant acce le ra tion period up to the knee point of the
ch arac te r is t ic . A period of constant power accele ration Is then required
above the c o rn e r speed of 55 K m /h r . This gives an output torque that is
inversely proportional to speed . The motor output power throughout this
period has to be 580 KW. In order to meet the braking requirem ents, the
braking torque at the motor shaft has to be 2 .2 8 KNm from maximum
speed to standstill. A gearbox efficiency of 0. 95 is assumed throughout.
116
3.7 KNm0
0
2.28 KNm broking
0
motoring0
2000 3000 50001000 6000
motor shaft speed1500I rpm)
1500
broking
1000
5 00 KW motoring
500
1000 2000 3000 5000 6000
motor stiof t speed1500I rpm)
A Constont Torque Accelerotion 0 - 5 5 K m / h r
B Const ont Power Accelerotion 55 - 2 2 5 K m / h r
Fig 3.2 T ra c t i o n Motor C t i a ra c t e r i s t i c s
117
T h e relative size and power supply requ irem ents of a particu lar m achine
design a re d epen dant upon the way in which it is controiied. Two methods
of control in co m m on use a re co ns idered . T h ese are:
1 A 'rising vo ltage' control sch em e
2 A constant vo ltage ' control sch em e
A constant output torque is required for acce le ra t io n and braking. This
type of ch arac te r is t ic is achieved under both control methods by
m ain ta in ing a constan t vo itag e -freq u en cy ratio. The rotor frequency of the
induction m ach in e is held constant during these periods, as is the field
cu rren t of the synchronous m ach in e . At ail times the synchronous
m ach in e is opera ted at a torque ang le of 90 deg rees .
Th e two control s ch em es differ In the method by which a constant power
a c ce le ra t io n period is produced. T h e 'rising voltage' control schem e
utilises a supply voltage that is varied in proportion to the square root of
the speed . This gives a stator curren t that d ecreases for increasing
s p e e d , for ail m ach in e types. For an induction m ach ine operating in this
m od e the rotor frequ ency is again held constant. Equivalent perform ance
is obta ined from a synchronous m ach in e , by varying the field current in
d irec t proportion to the a rm ature cu rren t. The 'constant voltage' control
s ch em e uses a supply voltage that is held nominally constant for a
constant power output. This gives a stator cu rren t that rem ains constant
for increasing speed . T h e Induction m ach in e is operated at a constant per
unit slip va lue , and the synchronous m ach in e with a field current that is
varied in inverse proportion to the speed .
118
3. 3 Design and p erfo rm ance predictions
As the curren t and flux densities are highest at the end of the constant
torque reg ion , this point is chosen as the design point. For m achine
designs of the type being considered , for use with ciass H insulation, a2
c u rren t density of 5 A /m m was chosen. H igher current density values of 2 36
up to 8 A /m m have been used in the design of traction motors . but
it is felt that m ach ines could not be designed with confidence at this
ra ting , without perform ing a detailed study of the heat transfer process.
For traction m a c h in e s , the use of the RMS curren t density over the duty
cycle is cons idered to be equivalent to a continuous rating. From an
analysis of the duty cyc le . Appendix 3 . 5 . 1 . and having set a continuous
cu rren t density ra ting , an estimate of the current densities required for
each part of the traction motor character is tic may be obtained. The
resulting cu rren t densities for motor operation under the two control
sch em es con s idered is shown in Fig 3. 3. Most electrica l steels are able
to support a flux density of 1 .4 Tesia without undue saturation or loss,
and there fore this value is taken to be the maximum allowable in each of
the following designs.
The main g eo m etr ica l constraint on the design of the stator is imposed by
the choice of outside d iam eter , which must be no g reater than 600 mm
to m eet the specification. This d imension is therefore fixed for ail
designs. A s im ple open rectangular stator slot is used. 50 mm in depth
and with a slot width to slot pitch ratio of 0 . 5 . Ail stator coils are
assum ed to be full p itched. Ail stator and rotor slots are assumed to have
a packing factor of 0. 5, with the exception of the squirrel cage rotor,
where in view of the c lose contact possible between the conductor and slot
119
Curren t Density
e.a
/ speed
4.17
2 3
STATOR
5 time (mins)
C urre n t
Densi t y 5 35
5 time IminsI
8.UCu rren t Densi ty
6. 63/s p e e d
A
t im e (mins)
C u rre n t Density
K D
1 2
ROTOR
A Rising Voltage Control B
8. 59K p ------- -" speed
2. 67
Constant Voltage Control
1 Constant Torque ( 0 . 5 6 6 mini
2 C o ns ta n t Power (4 .4 5 4 min)3 Balancing4 Constant Torque B rak ing ( 3 . 7 mins)5 S ta t ionary
Fig 3.3 Winding C u r r e n t Den s i t i es
120
t i mel mins)
a packing factor of unity Is assum ed. The only rem aining variable to be
specified for the Induction m ach in e Is the air gap length. In view of the
expected shock loadings to be Imposed on the shaft, the gap length Is set
to 3 m m.
A co m p le te sum m ary of the Input variables to the design process Is shown
In T a b le 3 . 2 . M ach in es labelled ' V a re for a rising voltage' operation
and m ach ines labelled '2' a re for operation under a constant voltage'
control sc h e m e .
The variation of the total m ach ine weight and length, ( Inc lud ing core and
end w indings, but excluding case d e ta i ls ) , for 2 , 4 , 6 and 8 pole designs
Is shown In Figs. 3 . 4 and 3 . 5 . The 4 pole design points for a rising
voltage control s c h e m e , are those whose com puter data Is reproduced at
the end of the previous chap ter to Illustrate the design method.
As the stator co re outside d iam eter Is fixed, the output power available ,
from a g eo m etr ica l point of view. Is a function of the core length. In each
design case shown , the core length has been adjusted to give the
required output power at the design point of 580 KW.
The rising voltage' control schem e allows the use of a h igher starting
cu rren t density for a given duty cycle RMS value. This enables a shorter
and therefore lighter design to be produced. M ach ine operation under a
constant voltage' control schem e gives a starting current density that Is
c lose to the duty cycle RMS value. This leads to the requ irem ent for a
longer core and consequently a heavier m ach ine .
As the pole num b er Is Increased the length of the stator and rotor end
windings Is reduced . This inactive part of the m ach ine , although essential
121
zok-oz>Qz
ON
zo»—o=)Qz
(T U)2 io z
§ Q X Z O ID Z O >• CtL lO
(N
IIQ Xz o Z) z o >.X in
i i
UJ_l Z <in in
CNUJ ing I s i—I Z < >- in
max. stator conductor current Kg density A /m m ^
8 U 5-53 8 44 5 53 8 44 5 35
m ax.rotor conductor curren t density A /m m ^ Kp
bUU 5-53 8-44 10-94 8-44 10-94
max. core f lu x density T B 1 4 1-4 1-4 1-4 1-4 1-4
synchronous speed rpm 1500 1500 1500 1500 1500 1500
s ta to r core outside dio.mmc^ 600 600 600 600 600 600
s ta to r slot depth mm dg 50 50 50 50 50 50
air gap length mm g 3 3 - - - -
slot width / slot pitch p 0 5 0-5 0-5 0 -5 0-5 0-5
coil pitch / pole p itch y 10 1-0 10 1-0 1-0 1-0
pole a rc / pole p itch - - - - 0 -7 0 -7
stator packing factor kppg 0 5 0 -5 0 -5 0-5 0-5 0-5
rotor packing factor k p p p 10 1-0 0 5 0-5 0-5 0-5
stator and rotor iron packing factor k(s . k
10 1-0 1-0 1-0 1-0 1-0
stator an d rotor c o n d u c to r re s is t iv i ty x10"°oh m M
2 0 2-0 2-0 2-0 2-0 2-0
stator a n d rotor conductor density x IO^Kg/M^ Ss U
0-8930 0-8930 0-8930 0-8930 0-8930 0-8930
iron d en s ity x IO ^ K g /M ^ Si 0-7871 0-7871 0-7871 0-7871 0-7871 0-7871
Table 3 .2 In p u t var iab les to the design process
122
o 0.
o
CT\O
>C7>
JDE3C
O
J0 ).30ÿ JdMOd MX sassoi
LT)a
Oc o □
men
sdjjduj ' m5uaI auuoi SSDUJ
123
l i s
<o
cw
<K>
E
x>C3
JO)0 O} J d M O d M X ■ sassoi cJCüE
3oooun
O O <3
CM
<O
□ O oo .
sdj|duj ' y |6ua| auuo) ' ssoui
124
to its operation , contributes nothing to the output. For an increasing pole
num ber the In fluence of the inactive end windings diminishes, and
co m p arab le m ach in e weights and lengths are produced for a particular
m ach in e type. For m achines operating under both types of control a
co ns iderab le reduction In weight is possible by Increasing the pole num ber
from 2 to 4 . Further reductions a re available by raising the pole number
further, but this is at the expense of h igher supply frequencies and shorter
slot p itches. The effect of the pole num ber upon the power factor Is
sim ilar In c h a ra c te r for m ach ine operation under both control schem es.
The round rotor synchronous m ach ine has markedly the poorest power
fac to r , whilst designs of the salient pole type give a power factor that is
essentia lly constant above 4 poles. The power factor of the induction
m ach ine is reduced for an increasing pole num ber, due to an Increased
leakage rea c ta n c e associated with the corresponding requirem ent for a
higher supply frequency .
A com parison of the total m ach in e losses, comprising of the total
conductor loss and an estimation of the core loss, (section 2 . 3 . 1 ) . is
also shown in Figs 3 . 4 and 3 . 5 . As would be expected, the induction
m ach ine incurs significantly less losses than e ither type of synchronous
m ach in e , due to Its g rea te r utilization of slot a rea .
In view of the above results and the desirability of keeping the supply
frequency as low as possible, the 4 pole designs seem best suited to this
particu lar app lication. Tab le 3. 3 gives the main dimensions of the 4 pole
designs, and Fig 3 . 6 shows the motoring and braking characteristics of
each of these m achines.
The supply VA capacity is de te rm ined by the product of the voltage
required at the highest operating sp eed , and the starting current. The
125
zoh-o3QZ
(Nzo»—oZ)Qz
q : LOiiQ X Z Oz> zs
CN
cr CO
2 g
§ g
|it r CO
LlJ CO
5 5_l z < >- CO CO
fN
g i_ J z
>-CO CO
core length w . 302 4 70 482 531 297 451
a ir gap g 3 3 33 31 3 7 42
slot pitch Ts 34 9 34 9 34 9 34 9 34 9 34-9
pole pi tch ^ p 314 2 314 2 314 2 314-2 314-2 314-2
depth stator backing core d . 50 50 50 50 50 50
s ta to r slots / pole phase 3 3 3 3 3 3
ro to r s l o t s / pole phase qp 3 3 9 9 1 1
s ta to r conductors per slot Zg 4 4 4 4 4 4
rotor conductors per slot zp 1 1 20 15 300 200
rotor winding factor kyyp 1 1 2/TT 2/TT 2:1 *1
pole body w idth Wp - - - - 100 100
nos. of vents n y 0 0 0 0 0 0
nos. of parallels for braking 2 2 2 2 2 2
all dimensions in mm
Table 3 . 3 Machine dimensions Diesel - e le c tr ic t rac t i on speci f icat ion
125
much la rg er VA produced in braking is accom odated by the use of parallel
connections and braking resistors, (section 2 . 5 ) .
127
§ii
s sA' » 6 o ) |0 A aso yd
V ■ lu a j jn o asoqd
VAH ' jn d jn o Xjddns
MW ' sso| JO)Sjsaj 6u!>)0jq
ceooz
z 6enO
3■O
A ■ aôot iOA a s o y d XouaiDijja ' j o p o j ja^od i ' X n s u a p x n n do6 jro
V ■ ( u a j j n o asoyd
128
z
I r i
A" a6o)|OA a so y d
V ' j u a j j n o a s o y d
VAW ■ j n d j n o X| ddns
/a n ■ s s o | J O ) S j s a j 6u!>)Djq
ÊenO
■oc
vO
m
A ' a6o*|OA esoqd X o u ap ! j /a ' j o p o / j aMod ' Xjjsuap xn|j do6 j i d
V ' ( u a j j n o a s D u |d
129
160
o 120
80
60
1600
1200
Iphase
600
field
20000 6000motor s h a l l speed
( r pm)
P. 300
200
100
< 1600
800
phase
600 field
2000
1.0 r
6000 5000m o to r S h a l t speed
I r pm)
£ 0.8
0.6
0 2
eff
Pf
Bg
2000 6000 hOOOmotor st'oft speed
I rpmj
Fig 3 .6c Synchronous Motor (Round Rotor) , Rising Vol tage Control - Motoring
130
160
120
80
iO
> 1600
1200
Ifield
800phose
0 2000motor shutt speed
1 rpm)
300
2 00
100
1600
1200
800
iOO
phase
field
2000 4000motor stiol! speed
('pm)
3.0
supply
2,0
resistor
020000
mn' til s peed
Fig 3.6d Synchronous Mofor (Round Rotor) , Rising Voltage Control - Braking
131
160 1600
o 120 > 1200phase
80080
field
20000 6000
1600
^ 300 1200
200 800
100 600
1 0
G 0.
y 0 6
0 6
X 0 2
5 0
2000
2000 6000
motor br.c ‘ f spee d1 r D m '
phose
f ield
6000 6000motor shc it speed
Irpm)
6■nolnr ■.(..'tl ' .ppcd
rpm;
Fig 3.6e Synchronous Motor (Round Rotor) , Constant Voltage Control - Motoring
132
160
120
80
to
1600
^ 12 0 0
bOC
too
field
phose
2000 tOOO 500Cmo for shaft speed
I rpm)
\ 1600
12003 0 0
8 0 02 00 f ield
too100
2000 tOOO 500Cmofor shaft soeed
I I pm I
0
0
resistor
0supply
02000 tOOO 6 0 0 0
motor shott jpeed ( r pm’
Fig 3.6f Synchronous Motor (Round Rotor) , Constant Voltage Control - Braking
133
160
120
8C
40
600
phase
field
200
0
300
200
100
1600
1200
800
400
1 . 0
£ 0.8
0 4
0 2
2000 4000
2000
2000
6000
motor shot# speed Irpm)
phose
field
4000 6000motor shoft speed
( r pm)
eff
Bg
4000 6000motor shaft speed
(rpm)
Fig 3.6g Synchronous Motor (Sal ient Po le) ,Ris ing Voltage Control - Motoring
134
160
120
80
40
8 Où
600
400
200
field
phose
2000 4000 6000motor shoft speed
Irpm)
300
2 00
100
1500
1200
800
400
phase
f leio
2000 4000 600Cmotor shoft speed
( rp m )
- 3
0
0
resistor
0supply
02000 4000 6000
motor shoft speed I rpm)
Fig 3.6h Synchronous Motor (Sal ient Pole) , Rising Voltage Control - Braking
135
160 eoc
o 120 > 600
coo
CO 200 field
2000 COOO
1600
F, 300 " 1200
200 800
100 coo
6000
motor s h a f t speed (rpm)
phase
field
2000 COOO 6 00 0m o fo r s h o f t speed
( rpm )1 . 0
£ 0 8
- 0 . 6
0 c
0 2
2000 COOO 6 0 0 0
motor shaf t speed (rpm)
Fig 3 .6 j Synchronous Motor ( Sal ient Pol e ), C onstant Voltage Control - Motoring
136
160
120
80
LO
> 600
600phase
I40C ( ield
200
2000 4000 6000motor shaft speed
(rpm)
300
1600
1200
100 400
phase
2 00 800 -
f ield
2000 4000 6000motar shaft speed
( rp m )
-
0
0
res is tor
0
SuppI y
02000 4000 6000
motor sh a f t speed I r pm)
Fig 3.6k Synchronous Mofor ( Salienf Pole) , Constant Voltage Control - Braking
137
3. 4 Conclusions
Tab le 3. 4 shows that even with the modest values of RMS current density 2
used of 5 A /m m . it is possible to design an AC m achine well within the
weight specification of 1600 KG. Even lighter designs may be possible, as
ultimately It is the tem p era tu re at which the winding can be safely operated
that will de te rm in e the maximum value of conductor current density. In
o rd er to d e te rm in e these limits a heat transfer study would have to be
undertaken .
The 'ris ing voltage' control sch em e produces lighter designs but incurs the
penalty of requiring a h igher supply capacity. Operation under a constant
voltage ' control has the benefit of requiring a sm aller supply capacity but
leads to a heavier design. This form of control is also associated with a
falling m ach in e power factor throughout the constant power region.
Of the three types of m ach in e considered the round rotor synchronous
a p p e a rs the least attractive. This is because of the large supply capacity
requ ired due to its inherently low power factor.
The induction and salient pole m ach ines a re sim ilar in both physical size
and p e rfo rm an ce , and requ ire supplies of a similar capacity. However
b e c a u s e of the la rge air gap . both types of salient pole machine are
approxim ate ly 25% lighter than the corresponding induction machine. This
ap p e a rs to make the salient pole synchronous m achine the most suitable
for this particu lar app lication , but the following points must be weighed
against it.
138
k :>>CL
T3 CL
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139
1 The contro l system requ ired to keep the torque angle constant is
re latively com plex.
2 The cost of a s e p a ra te field supply.
3 The use of slip rings.
4 A less rugged and m ore costly rotor construction .
In the fo llow ing c h a p te r, the p erfo rm an ce of the lightest designs of
Induction and synchronous m ach ine Is Investigated , when being fed by
n o n -s ln u so ld a l supp lies with d iscontinuous phase curren ts . Two Inverters
are c o n s id e re d , one of the constan t vo ltage , and one of the constant
cu rren t type.
In o rd er to furn ish the equivalent c ircu it p aram eters necessary for this
Investigation , a fu rth e r com puter program has been written to enab le
these p aram ete rs to be ca lcu la ted from the physical m achine d im ensions.
The p a ra m e te r ca lcu la tio n s a re based upon the m ateria l presented In
Appendix 2 . 7 . 2 . p rio r to any sim plifying assum ptions being m ade for the ir
use In the design p rocess . The resulting p aram eters for the two m achines
are as follows:
Induction m ach in e Fig 3. 7
Stator res is tan ce
Stator leakag e In d u ctan ce
R eferred rotor res is tan ce
R eferred rotor leakag e Inductance
M agnetis ing In d u ctan ce
M otor w eight
7. 015 m fl
0 .1 8 7 3 mH
2. 404 mfl
0 .1 9 9 1 mH
5. 879 mH
787 KG
UO
Synchronous m ach in e Fig 3. 8
Stator res is tan ce 6 .9 7 7 m n
Stator leakag e Ind u ctan ce 0 . 1851 mH
R eferred fie ld w inding res is tance 1 .9 9 8 m fl
D irec t axis m agnetis ing Inductance 0 .4 8 7 4 mH
Q uad ra tu re axis m agnetis ing Inductance 0 .2 2 5 2 mH
M otor w eight 624 KG
A rising out of the m ach in e design work presented In this chap ter a re a
num ber of a re a s In which fu rth er study would be b enefic ia l. Firstly the
co m p u ter m ethods p resented should be checked against practical results
from su itab le h ighly rated m ach ines .
Second ly In o rd e r to establish w hether h ig h er values of RMS cu rren t
density could be used , a study of the heat tran s fer ch aracteris tics of AC
m ach ines should be u ndertaken . If the use of h igher values of cu rren t
density Is feas ib le fu rth er reductions In m ach ine weight would be possible.
F inally as the s im plified design fo rm ulae used to com pare the d ifferen t
m ach in e types use the m inim um num ber of variab les , a sensltklty analysis
could easily be Incorpora ted Into the design m ethod. This would lead
natura lly to the use of optim isation techn iques In com pleting a design.
U1
900
6 263 0 2
air gap
321600
oil d im ens ions in mm
A5
18 32
26 18
A
16 75
63
5 5
S T A T O R SLOT RO T O R S L OT
Fig 3.7 Induct ion Motor T rac t ion Speci f icat ion
U 2
900
6 2 ' 2 97
20 50
Qir gap
100 326 600600
polewidth
Z J .
oil dimensions in mm
5
18 32
26.18
STATOR SLOT
Fig 3,8 Synchronous Motor (Sa l ien t Pole) T rac t ion Speci f icat ion
U 3
3. 5 APPENDIX
3. 5. 1 The traction motor duty cycle
D ISTANC E (K M ) STATION TIM E (m in u tes )
FROM INCREfVlENT INCR EM EN T STATION ACCUM .
ORIGIN DWELL TOTAL
0 . 028 . 1
LONDON. EUSTON13. 7
- 0. 0
28. 1
104. 7W ATFORD
3 4 . 41.0 14. 7
132. 823. 5
RUGBY10. 0
1.0 50. 0
156. 358 . 7
NUNEATO N20. 6
1.0 6 1 .0
2 14 . 939 . 4
STAFFORD15. 9
1.0 82. 6
2 54 . 338 . 8
CREWE15. 8
2 . 0 99. 4
293 . 118. 9
W ARRINGTON8 . 9
1.0 117. 2
312 . 124 . 3
WIGAN10. 7
1.0 127. 1
336 . 433 . 8
PRESTON13. 4
2 . 0 139. 8
37 0 . 21 1 1 . 3
LANCASTER42. 3
1.0 154. 2
4 8 1 . 6118. 6
CARLISLE42. 4
2 . 0 198. 5
6 00 . 1
25 . 5CARSTAIRS
1 1 . 51.0 2 4 1 . 9
625 . 620 . 6
MOTHERW ELL11.6
1.0 2 54 . 4
646 . 3 GLASGOW CENTRAL_. ...
- 266 . 0
13 a c c e le ra tio n periods under m otoring ch arac te ris tic Fig 3 .2
13 d e c e le ra tio n periods under braking ch arac te ris tic Fig 3 .2
15 m ins when m otor curren ts a re zeroTrain o p era tes at 225 K m / h r at all o ther tim es so no account is
taken of banking or other form s of reduced speed operation .
U 5
C H A PTER 4
TH E PER FO R M A N C E OF INVERTER FED AC M ACHINES WHOSE
PHASE C U R R EN T IS D ISC O N TIN U O U S
4. 1 Introduction
In th is ch ap ter a co m p u ter m ethod Is presented that enab les the steady
state p erfo rm an ce of induction and slip ring synchronous m achines to be
evaluated w hen the phase cu rren t becom es d iscontinuous.
T he operation and analysis of two types of inverter is considered and
co m p u ter p red ictions a re presented to dem onstrate the perfo rm ance of the
induction and sa lien t pole synchronous m otor designs, of C hapter 3 .
The two types of in verter considered are:
1 V o ltage S o u rce In verte r (V S I)
2 C u rren t S ource In verter (C S I)
Both Inverter types a re operated in the 120 d eg ree conduction m ode.
For a voltage source inverter operating in the 120 d eg ree conduction
m o d e, only two output thyristors a re gated on at any one instant, and
each thyristor will conduct for 120 deg rees of the output period. Logic
c ircu itry and gate pulse in form ation , can in this c a s e , be sim plified as a
thyristor is not gated into conduction until 60 deg rees after Its
com plim entary thyristor has been turned off. However for high power
fac to r loads, which is the case for the two m otor designs under
c o n s id e ra tio n , the phase cu rren t is unable to reverse d irection in the 60
d e g re e period follow ing thyristor turn off. During this zero curren t interval
the m ach in e back em f appears at the inverter output te rm in a ls . Thus for
inverter operation in the 120 d eg ree m ode, the inverter output voltage is
a function of the m ach in e param eters and the loading condition .
U7
W hen m odelling a cu rre n t source inverter, it is frequently assum ed that
the DC link c u rre n t is constan t. This req u ires an infinite value of link
in d u c ta n c e , w hich in p rac tice is unrea lis tic . Additional approxim ations31
have a lso been m a d e . by neg lecting the stator res is tance and
assum ing the m ach in e back em f to be constan t during the com m utation
period .
An exact m odel of the C u rren t S ource Inverter Is presented h ere . In which
the e ffec t o f th e DC link inductance is taken into acco u n t, by considering
the in verte r to be supplied from an ideal voltage source. The foregoing
approxim ations reg ard in g stator res is tance and m ach ine back em f a re not
m ad e in this study, and the m ach ine m odels em ployed use all the norm al
equ iva len t c irc u it p a ra m e te rs .
B ecau se of the thyristor switching sym m etry in h eren t in both the VSI and
C S I. it is n ecessary to cons ider only one sixth of a cycle of inverter
o p era tio n . A steady state solution m ay then be determ ined for any
o p eratin g po in t, by the in tegration of the system equations over this 60
d e g re e p e rio d , and equating the in itial and final values. A com plete
solution for one cyc le of inverter operation m ay then be constructed by
sym m etry , f rom the 60 d e g re e solution.
An induction m ach in e m odel is p resented in this ch ap ter, in which rotor
variab les a re tran s fo rm ed to the stator, thus rem oving the dependency of
the e lem en ts of the m ach ine m atrix on the rotor position. This
transfo rm ation has advan tages over a d -q representation of the induction
m a c h in e , in th is ap p lica tio n , because as the actual th ree phase currents
are a v a ila b le . the ch an g es of state within the inverter a re readily
d e te c ta b le , w ithout the need to transform a two axis system of currents
back into a th ree phase system .
U 8
M a c h in e vo ltage and cu rren t w aveform s are shown for the VSI and CSI fed
induction and synchronous traction m otor d es igns , and a com parison is
m ad e of the h arm o n ic conten t p resen t in the output torque w aveform s.
T h ese m ach in es a re operated under a rising voltage control sch em e,
d e scrib ed in the previous c h a p te r, to give an output charac te ris tic that
m eets the req u irem en ts of a high speed d ie s e i-e ie c tr ic traction m otor.
U9
4 . 2 M ach in e M odels
4 . 2 . 1 induction m ach in e
The norm al d irec t th re e -p h a s e represen ta tion of the induction m ach ine
gives an ind u ctan ce m atrix , in which the e lem ents are a function of the
ro tor position. T h e p e rfo rm an ce equations a re there fo re d ifferentia l3 7 . 3 8
equations with v a riab le co e ffic ien ts . An induction m ach ine model
has been deve lo p ed , that is based upon a coord inate system that is fixed
in the stator. This m odel is d escribed fully in Appendix 4 . 8 . 2 . As the
transfo rm ed rotor cu rren ts a re at the sam e frequency as the sta to r, the
tim e taken to eva luate a steady state solution is reduced . A lso, as the
th re e phase vo ltages and cu rren ts a re used, the in v e rte r-m a c h in e
equations a re ab le to be set up with re lative ease . The induction m achine
d ifferen tia l equ atio n s . Fig 4 . 1 . a re p resented in the following fo rm , in
m atrix notation
v = ( R + w G + L p ) i ( 4 . 1 )R
w here the vectors v and i re p resen t the phase and re ferred voltages and
c u rren ts respective ly . For e a s e of presentation the inductance m atrix L.
im p ed an ce m atrix G and d iagonal res is tan ce matrix R have been
com b in ed . Note that the com ponents of im pedance matrix G do not vary
with the rotor position.
The perfo rm an ce equations of Fig 4.1 contain coeffic ients that are
d ependant only upon the norm al equ iva len t c ircu it values and the steady
rotor an g u la r velocity, w .R
150
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151
Inverters of the type to be considered produce an output current waveform
that has a high harm o n ic con ten t. This leads to the production of39
harm on ic torques that tend to induce speed oscillations of the rotor,
it is assum ed for the purposes of this study that the rotor has sufficient
in ertia to m in im ise this e ffe c t, and thus revolves at a constant speed.
in the following an a lys is , core losses, M M F space harm onics and
m agnetic sa turation effects have been n eg lec ted , and ail m achine
p aram eters a re assum ed constan t and independant of frequency. Both the
Induction and synchronous m ach ines a re star connected .
The e lec tro m ag n e tic to rque developed by an Induction m otor having p pole
pairs is given by
TT = e i ^ i ( 4 . 2 )
e 2 d e
Tw h ere i is the transpose of i. This equation becom es,
from appendix 4 . 7 . 2
T = - p j 3 Lm [ i ( i - i ) + i ( -1 +i ) +i ( I - i )]e 3 aS bR' cR' bS aR' cR' cS aR' bR'
(4 . 3)
The phase and re fe rre d cu rren ts of equation ( 4 . 3 ) a re the instantaneous
values.
152
4 . 2 . 2 Synchronous Machine
The synchronous m a c h in e m odel is shown in Fig 4 . 2 . and is based upon
m ateria l p resen ted in re fe re n c e 40 . The stator Is assum ed to be wound
with a ba lan ced se t of th re e phase co ils , and the rotor has a s ing le field
co il. T h e re a re no d and q axis d am p er w ind ings. All space harm onics
of M M F and flux density above the fundam enta l a re considered to be
neg lig ib le . Coil F ' rep resen ts the fie ld w inding and is fed via slip rings
from a DC supply.
For the coil system of Fig 4 . 2 , the instantaneous voltages of the m ach ine
in te rm s of flux and c u rre n t a re given by the m atrix equation
V = Ri + p<D (4 . 4)
w here <D = Li
This expression m ay be expanded to give
V = Ri + w Gi + Lpi (4 . 5)R
w here the matrix G = ^ and the ro tor speed w = ^d e R dt
The vo ltage vector v represen ts the stator phase and re ferred field
vo ltages.
TV = [ V V V V ' J
aS bS cS F
153
bS
oS
'cS
"aS
Ng - e f f ec t i ve s t o t or t u r ns per phase
Np - " t u r n s of f ie ld w i n d i n g
L md = 3 N s^ Lmq = 3 N s
2 ”R d zR,
F ig ^ . 2 S y n c h r o n o u s Ma ch ine Model
Vector I represents the phase and referred field currents,
TI = [ I I I I ' 1
aS bS cS F
and R = d iag o n a l t R R R R 1S S S F
The synchronous m ach in e Inductance m atrix L Is shown In Fig 4. 3 and Is
ap p licab le to both sa lien t pole and round ro tor m ach ines. In the case of
the round ro tor m ach in e Lmd = Lmq. The e lem en ts of the L m atrix a re
d ep en d an t upon the Instan taneous position of the rotor at any tim e, t. The
an g u la r position of the ro tor with re fe re n c e to the a ' phase axis Is given
by.
0 = a > t + ' y - 7 t “ 6 ( 4 . 6 )R 2 T
w h ere 0 Is the to rque an g le . The an g le y corresponds to the e lec trica l T
phase d isp lacem e n t of the fundam ental com ponent of the motor cu rren t.
with resp ect to the com m utation point t . Fig 4 .5 and 4 . 1 0 . at the starto
of the 60 d e g re e period over which the m ach ine equations a re to be
In teg ra ted .
The e le c tro m a g n e tic to rque developed by a synchronous m otor having p
pole pairs Is given by. In matrix notation
TT = Ê I G I ( 4 . 7 )
e 2
The Instan taneous value of output torque Is readily availab le by
considera tion of the In tan taneous phase and re fe rred rotor curren ts , and
the m atrix G.
155
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156
4 .3 V o ltage S ource Inverter: O perating states and form ulation of system
equations.
The vo ltage so u rce Inverter considered In the following analysis Is shown
In Fig 4 . 4 . Thyristors T1 to T 6 a re gated sequentially accord ing to the
switching pattern of Fig 4 . 5 , and rem ain in the conducting state for 120
d eg rees of the ouput period. Diodes 01 to 0 6 enab le the reactive load
en erg y to be c irc u la te d , and allow reg en era tio n back Into the OC link if
re q u ired . T h e OC so u rce vo ltage. Vs. Is assum ed to be Ideal.
The tim e Interval T . Fig 4. 5 . chosen for the analysis of the system
p erfo rm an ce is the 60 d eg ree period In itiated by the firing of thyristor T l .
at tim e t . During the tim e Interval T . two distinct c ircu it states exist, o
State A - all th ree m ach in e phases a re connected to the inverter
State B - m ach in e phase 'c ' Is d isconnected from the Inverter
(F ig 4 . 6)
P rio r to tim e t . thyristors T5 and T 6 have been conducting and m achine o
phase 'a ' has been d isconnected from the inverter output term inals . At
tim e t . thyristor T l is gated on and thyristor T5 will be extinguished, o
D iode 0 2 is now connected across the m ach ine term inals B and 0 . and
is b iased in such a way as to perm it conduction . The equations describ ing
this opera ting state a re obtained by considering the two conduction loops
of Fig 4. 6 . which is equivalent to. For State A
Vs = v - VaS bS
(4. 8)
Vs = V - V
aS cS
157
Vs
03T 1
0604 T 6 02 T2
Fig 4 .4 Vo l tage Source I n v e r t e r
- O Q
-O b
-O c
60 360
Tl Tl m T4 T4
T6 T3 T3 T6
w T2 T2 m T5 T5
t . T L Diode conduction period
(load dependent )
Fig 4.5 Thyr i s to r Swi tching Sequence
158
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atn
159
noting that the m ach in e phase a' current is given by
i = - ( I + i ) ( 4 . 9 )aS bS cS
The system will con tinue to o perate in this m ode until the diode current
I has fa llen to ze ro . This conduction period Is a function of the load D2
p a ra m e te rs . W hen d iode D2 ceases to conduct, m ach ine phase 'c ' will be
d isco n n ected from the inverter. The sytem is now in State B. and is
d escrib ed by the fo llow ing equations.
Vs = V - V ( 4 . 1 0 )aS bS
w here the m ach in e phase currents a re given by
I = - ibS aS
(4 . 11)
and i = 0cS
S tate B continues for the rem a in d er of the 60 degree period under
observation and ends with the gating on of the next thyristor in sequence,
which in this case is thyristor T2.
The In v e rte r -m a c h in e system equations may now be d eterm ined , by the
substitution of the app ro p ria te m odel of the induction or synchronous
m ach in e into the above expressions for source vo ltage, ( 4 . 8 , 4 . 1 0 ) ,
taking into account the phase curren t re lationsh ips, ( 4 . 9 , 4 . 1 1 ) . that
exist in the two c ircu it states. The d ifferen tia l equations representing State
A and S tate B. Figs 4 . 7 and 4. 8 . a re in tegrated over the period T by
d ig ita l co m p u ter m ethods. The point at which the circuit experiences a
c h a n g e of s ta te , is found by m onitoring the m agnitude and direction of the
curren t in diode 0 2 .
160
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K l = e K 2 : 1 8 . el K 3 = l e . 2 t )
Vs
Vs
- 2 R ^ . 2^w^L b! sin 2 Kl
- 2 s i n 2 K2 . s l n2 K3 1
- pi 21g ♦ Lo ♦ Lbicos 2 Kl
. 2cos 2K2 . C OS 2 K3 11
- Rc * 2w L b I 2s i n 2K15 J R
- s i n 2 K 2 - si n 2 K 3 |
- pl lg ♦ La ♦ Lbl2cos2K12 3
-COS 2K2 - CCS 2K 3 )|
w ^ y ^ L m d l sin K3 - sin Kl |
♦ p y^ Lmd [cos Kl - cos K3 1
11.2)
- 2Rc ♦ 2w L b i s i n 2K13 R
« sin 2 K 2 - 2 s in 2 K3
- p [ 2 l g . L o . Lb(cos2K1
♦ COS2K2 - 2cos2K311
v^y2 Lmd | s i nK2- sinKll
. p y i L m d [cosKI - cosK21
- ( 1 . 3 ) - 1 2 . 3 1 Rp » p( Ip ♦ L md )
bS
'cS
' F
Sf Q t e A
Vs
2Rc - 2 w L b i s m 2 Kl S y R
- 2 sin 2 K2 . sin 2K 3 ]
♦ p ( 2 l g ♦ Lo « L bfcos 2K1
- 2cos 2 K 2 . c o s 2 K 3 |
L md 1 sin Kl - sinK3l
«p y^2 Lmd (cos Kl - cos K3 |
(1.2) Rp . p l lp ♦ Lmd )
q S
S t a t e B
Fig 4.8 Vol tage Source I n v e r t e r Fed Synchronous Motor
163
W hilst the co m p u ter m odel developed for the VSI has been prim arily
designed to solve fo r discontinuous phase cu rren ts , if the load is
suffic iently inductive the curren ts will not becom e discontinuous. In this
case the c ircu it will rem ain in State A and the integration procedure will
be rep eated as b e fo re . Thus a steady state solution for the 180 d eg ree
conduction m ode m ay be found, producing the ch arac te ris tic q u a s i-s q u a re
te rm ina l vo ltage w aveform associated with this m ode of operation .
164
4 . 4 C urren t S ource Inverter: O perating states and form ulation
of system equations
The c u rren t source inverter Is shown in Fig 4. 9. The curren t source Is
m odelled by an ideal voltage source in series with a link inductor. Ldc.
w hich has to be ra ted at operating cu rren t levels.
Thyristors T l to T 6 switch the DC link c u rre n t, Id c , at a rate necessary
to estab lish the d es ired m ach ine operating frequency . C apacitors C l to C6
provide the en erg y storage requ irem ents for satisfactory thyristor
com m u ta tio n , and the diodes D1 to D 6 iso late the capacito rs from the
output te rm in a ls except during com m utation .
During co m m u tatio n , the capacito rs becom e charged to peak voltages
which a re large ly determ in ed by the level of m ach ine current being
co m m u tated . This en ab les the inverter to com m utate satisfactorily over a
wide ran g e of output frequency and vo ltage.
The 60 d eg ree p e rio d , T , for which the system Is to be observed,
co m m en ces at tim e t , Fig 4 . 10. During this period the DC link curren to
is tran s fe rred from phase a ' to phase b '. For each com m utation period ,
T , the inverter c ircu it has th ree distinct operating m odes, these are
S tate A - charg ing m ode
State B - cu rren t tran s fer m ode
State C - norm al m ode
165
Rdc Ldc
Vs
T3 T5C3
C5
0 3 05
06 02C2
Ci C6
T6T i
-O o -O b- O c
Fig 4.9 Current Source I n v e r t e r
50 3 6 0
Tl Tl T4 T4
T6 T3 T 3 T6
T2 T2 T5 TS
Fig 4.10 Th y r i s to r Swi tching Sequence
166
It Is assum ed that no com m utation overlap o ccu rs , le com m utation of the
phase curren ts Is successfully com pleted within the 60 degree period.
P rio r to the c o m m en cem e n t of the ch arg ing m ode, at time t . thyristorso
T l and T2 have been conducting . Fig 4 .1 1 a . At tim e t . thyristor T3 Iso
gated on and the c h a rg e stored across the com m utation capacito r C eq .
Is p resen ted across thyristor T l . which Is extinguished. As the tim e taken
to successfu lly com m utate thyristor T l Is very sm all com pared to the total
com m utation p e rio d , the transfer of cu rren t from T l to T3 Is assum ed to
be Instan taneous. During State A . Fig 4 .1 1 b . the delta connected
c a p a c ito r bank C l . 0 3 and 0 5 (fo rm in g the equivalent capacito r Oeq ) .
Is ch arg ed linearly by the constant link cu rren t. Idc.
In itia lly the m ach in e term inal voltage . v , Is m ore positive than theba
vo ltag e , v . across the equivalent c a p a c ita n c e , so reverse biasing the ceq
series d iode D 3. which th ere fo re does not conduct. The equations
d escrib ing this m ode of operation a re obtained by considering the
conduction loop of Fig 4 .1 1 b .
For S tate A
Vs = ( Rdc + pLdc ) I + V + V - v ( 4 .1 2 )aS ceq aS cS
and Oeq p v = 1ceq aS
noting that the m ach in e phase curren ts are
Idc = 1 = - IaS cS
and I = 0 bS
( 4 . 1 3 )
167
IdcRdc Ldc
Idc
ceq
D3Ceq
VsVs
02
03Ceq
'aS
'cS
02
T2
o) Circuit prior to time tg b) Charging Mode S TA T E A
I dc Id c
Vs
Ceq T3Ceq
c) T ra n s fe r M o d e STATE B d) N o rm a l M od e S T A T E C
Fig 4.11 Commut a t i on Modes of the C u r r e n t Source I n v e r t e r
168
The c h a rg e on the equivalent c a p a c ito r, C eq . reverses until v = vceq ba
and the series d iode 0 3 becom es forward b iased . As soon as diode 0 3
starts to co n d u ct, the 0 0 link Is p resented with a paralle l path through the
outgoing phase a ' and the Incom ing phase c '.
The In verte r c ircu it Is now operating In the transfer m ode. In which the
link c u rren t rap id ly transfers from phase 'a ' to phase b '. Fig 4 .1 1 c .
Ouring c u rren t tran s fer the m ach in e phase resistance and leakage
In d u ctan ces form a part of the com m utating c ircu it together with the
equ iva len t com m utation c a p a c ita n c e , and hence the variation of m otor
c u rre n t will follow a dam ped sinusoid . At all tim es during this m ode the
sum of phase 'a ' and b' cu rren ts will equal the 0 0 link value. The
equations d escrib in g this m ode, for the two conduction loops of Fig 4 . 11c
a re . for S tate B
Vs = ( Rdc + pLdc ) ( I + I ) + V - V (4 . 14)aS bS bS cS
Vs = ( Rdc + pLdc ) ( I + I ) + V + V - v ( 4 .1 5 )aS bS ceq aS cS
and Ceq p v = 1 (4 . 16)ceq aS
w here the m ach in e phase c ' cu rren t Is given by
I = - ( I +1 ) (4 . 17)cS aS bS
The m ain c h a ra c te r is tic of the tran s fer m ode Is the production of voltage
spikes across the m ach ine output te rm in a ls . The voltage spikes are
g en e ra te d by the rapid ly chang ing curren ts In the m achine leakage
Ind u ctan ces .
169
State B con tinues until the phase a ' cu rren t reaches zero . When this
o ccu rs , the series d iode D1 ensures that the curren t cannot reverse
d irec tio n , and thus iso lates m ach ine phase 'a ' from the inverter output
te rm in a ls . C om m utation cap ac ito r 01 Is now charged to the correct
polarity for Its next com m utation duty.
C urren t tran s fe r is now com plete and the inverter enters the norm al m ode.
State O. Fig 4 . l i d . During this m ode the com m utation capacito r voltages
rem ain co n stan t, and the DO link c u rren t flows through m achine phases
b' and 'c '.
For State O th e re fo re .
Vs = ( Rdc + pLdc ) i + v - v ( 4 .1 8 )bS bS cS
and the m ach in e phase curren ts are
- I = I and I = 0 (4 , 19)cS bS aS
This state con tinues for what rem ains of the period T . after which a
fu rther com m utation period is in itiated by the gating on of thyristor T4.
The co m p le te in v e rte r-m a c h in e equations may now be set up in a sim ilar
m an n er to the vo ltage source Inverter system , by the substitution of the
a p p ro p ria te m ach in e m odel Into the above expressions for the three circuit
states. The resulting equations for the OSI fed induction and synchronous
m ach ines a re shown in Figs 4. 12 and 4. 13. respectively.
These d iffe ren tia l equations are evaluated for each mode of operation,
over the period T. The two changes of state are found by observing the
170
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176
point at which the vo ltage across D3 goes positive and the current through
D1 has red u ced to zero .
If Ins tan taneous com m utation Is assum ed the phase curren t waveforms are
re c ta n g u la r, in 120 d e g re e blocks. In this case the DC link current is
given by
Id c = TLJ t lp h ( 4 .2 0 )^ rms
This equation gives a good approxim ate value for the DC link curren t,
and Is used as the starting value In the Iterative process to evaluate a
steady state so lu tion .
An In itia l value for the voltage across the com m utation capac ito r C l . for
a star co n n ec ted m ach in e and neg lecting the m ach ine back em f. Is given 31
by
V = V = 2ldc ( 4 .2 1 )c l ceq 3Cw
c
w here w = JLc V 3L C
T
and L Is the total m ach in e leakage Inductance and C . the Inverter T
c a p a c ita n c e va lue . The appropria te value of source voltage. Vs. at any
m ach in e o p era tin g torque m ay be found by equating the DC link power and
the total m ach in e Input pow er.
177
4 . 5 C om putational P rocedure
The d iffe ren tia l equations that d escrib e the In v e rte r-m a c h in e perfo rm an ce ,
m ay be solved by any num erica l In tegration procedure using a step by
step m eth o d , providing the step length is sm all enough. In genera l two
m ethods a re ava ilab le
1 m ultlstep m ethods - In which the know ledge of a previous solution Is
used at each In tegra tion step for the ca lcu la tion of future points.
2 s ing le step m ethods - In which all evaluations of the algorithm are
confined to a s ing le In tegration step and previous solutions are not
re q u ire d .
P re d lc to r-C o rre c to r m ethods belong to the first group, w hereas
R u n g e -K u tta m ethods belong to the second . A lthough a cho ice between
th ese m ethods tends to be difficult to m ake , the genera l considerations
w hen se lec tin g an a lg o rith m , apart from the nature of the problem to be
solved a re . num erica l stability, sim plicity of the algorithm In term s of its
Im p lem entation on the co m p u ter, and the tim e and m em ory requirem ents.40
A R u n g e -K u tta -M e rs o n In tegration m ethod . Is em ployed throughout
this Investigation , for the following reaso n s .
1 no sp ec ia l starting procedure Is req u ired , which has advantages during
trans ien ts and w here the num ber of equations to be solved Is changing .
2 the step length Is easily changed .
3 a s tra igh tfo rw ard com putational p rocedure Is repeated throughout the
in tegration process and gives accu ra te resu lts .
178
An estimate of the accuracy of the method is given by the computation of
an e rro r fu n ctio n , from the w eighted sum of the individual estim ates at41
ea c h In teg ra tio n step . A su itab le step length was chosen to keep this
e rro r w ithin re a s o n a b le lim its.
T h e system equations a re evaluated for one sixth of a cycle only. The
In teg ra tio n p ro ced u re is then repeated fo r successive 60 degree periods
until a s teady state solution is rea c h e d . At the end of each period , T , the
system v a ria b le s , e g . the phase and re fe rred rotor curren ts , and
com m utation c a p a c ito r voltage (In the case of the curren t source
In ve rte r) . a re red es ig n ated and used as the new starting conditions for
the next In tegra tion period . W hen the In itial and final values for one sixth
of a cyc le have settled to within a given to le ra n c e the system is assum ed
to have re a c h e d the steady state. The 60 d e g re e solution is then used to
co n stru c t the m ach in e p erfo rm an ce w aveform s for one com plete cycle of
Inverter op era tio n .
Iterative loops a re provided within the com puter program to accurate ly
d e te rm in e the point at which a ch an g e of state occurs . The step size is
progressive iy red u ced untii the variab le under observation has reached the
desired value to within a given to le re n c e .
All co m p u te r p rogram s w ere written using the FORTRAN language and all
ca lcu la tio n s w ere perform ed on a Honeywell 68DPS com puter at the
University of Bath.
179
4 . 6 P e rfo rm an ce pred ictions for Induction and sa lien t pole synchronous
m otors.
S teady state p e rfo rm an ce predictions fo r the Induction and salient pole
synchronous designs of C hap ter 3 , a re shown In Figs 4 . 16 and 4. 17. for
the vo ltage and cu rre n t source Inverters respectively. T h ree sam ple
freq u en cy points a re show n. 10, 50 to 2 00H z. to dem onstrate the
m ach in e responses over the requ ired speed ran g e .
Both inverters a re operated under a rising voltage contro l. In each case
the In verter so u rce vo ltag e . Vs. has been set to give the output torque
req u ired by the trac tio n m otor ch arac te ris tic of Fig 3 .2 ,
A h arm o n ic analysis of the output torque waveform has been perform ed to
d e te rm in e the m agn itudes of the 6th and 12th harm onic torques present.
T h ese two harm onics d om inate , due to the nature of the switching
s eq u en ce in the in verter. The resuits of this analysis can be seen in Tab le
4 .1 . T h e p ercen tag e figures quoted rep resen t the ratio of peak harm onic
to rque to averag e torque levels.
It can be seen from Fig 4 . 16 that the m ach ine phase voltages, for a VSI
fed m ac h in e , a re d ep en d an t upon the m otor back em f during the periods
for w hich one of the m otor phases Is d isconnected from the inverter. At
all o th er tim es the te rm in a l voltage Is well defined . Tab le 4. 1 shows that
the m agn itude of the harm onic torques p resen t in the output of VSI fed
m ac h in e s , a re co n s id erab ly reduced for the synchronous m ach ine. This
would suggest that a VSI fed synchronous m ach ine would be preferab le in
this ap p lica tio n , but the sam e restrictions regard ing the com plexity of the
ro tor position contro l system , and the cost of a separa te field supply must
again be w eighed aga inst Its reduced to rque ripple capabilities .
180
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181
Th e DC link In d u c tan ce of the C S I, Ldc, has been set to ten tim es the
total le akag e In d u ctan ce of the induction m otor. Ldc = 4 m H. For
com p ariso n purposes this value was unaltered for determ in ing the
p e rfo rm a n c e of the sa lien t pole synchronous m otor. The link res is tance .
R dc. was set to ze ro . A com m utation ca p a c ito r of 10 /iF was used as this
value successfu lly com m utates the link cu rren t at the upper frequency lim it
of the system , with no com m utation overlap .
The resu lting phase voltage waveform of the CSI fed m achines Is
s in u so id a l, with superim posed com m utation spikes due to the rapidly
chan g in g cu rren ts In the m ach ine leakage Inductances. The m agnitude of
the phase vo ltage Is dep en d an t upon the fundam enta l cu rren t com ponent,
and the o p era tin g point of the m ach in e . O ne advantage of the CSI. Is that
this vo ltage is useful for control purposes with a m inim al am ount of
filte rin g .
The com m utation vo ltage Is also a function of the m otor cu rren t, so that
as the m otor c u rre n t in creases due to a ch an g e In the operating point,
the ability of the Inverter to com m utate that cu rren t is increased
a cco rd in g ly . This m eans that the la rgest com m utation spikes occur during
the high c u rre n t req u irem en ts of the constan t acce le ra tio n period , up to
a tra in speed of 55 K m /h r . This fact Is illustrated In Fig 4 . 17. As a result
of the In c rease d vo ltage stress at low operating frequencies a carefu l
cho ice of insulating m ateria ls will be requ ired to prevent long term
d e te rio ra tio n of the w inding Insulation.
C om paring the m agn itude of the harm onic torques present in the output of
the CSI fed m ac h in e s , suggests that the induction m otor design would be
m ore s u ita b le , as the p e rcen tag e torque ripp le present In the output Is
co n s id erab ly low er than for the synchronous design.
182
T h roughout this analysis the equ iva len t c ircu it p aram eters have been
assum ed to be In d ep en d an t of frequency . This Is only approxim ately so,
as skin e ffec ts will lead to an In c rease In ro tor res is tance and a d ecreas e
In le akag e re a c ta n c e as the frequency In c rease s . How ever this factor will
have m in im al e ffec ts upon the m odel, which It Is fe lt produces a
suffic iently a c c u ra te Ind ication of the m otor te rm in a l ch arac te ris tics , to
allow fo r th e re a lis tic rating of the Inverter com ponents .
183
PHSSr CURRENTX»02
PhASr voltage VOSX)0-
10.
0
XI0-5:
mS10
- 2 .
mS
LI NE V O L T A G E vob TORQUE
X10-I 0_,
0 - ,10
X10
mS
X10
10mS
F ig A 16a V S I I n d u c t i o n M o t o r 1 0 H z
ISA
PHASE CURRENT ios PHASE voltage /a?X102 XI02)5^
20
-5:
- 10 :
•J5.1
mS20
- 2 :
- 4
mS
LIN E VOLTAGE vobTORQUE
XI 0210-,
201510
-5.
10."
mS
X1025
mS2010
Fig Z1.16b VSI I n d u c t i o n M o to r 50 Hz
185
PHASE CURRENT lo i PHASE voltage v o s
X10“ XJ0215^
- 5
-10
mS
10,
o>mS
-5
L IN E v o l t a g e vobTORQUE
X)0
fr.S0 - .
-2J
X1035 ,
1.
_» mS0 1
F i g ^ . 1 5 c V S I I n d u c t i o n M o t o r 200 Hz
186
PHASE CURRENT iasX I02 XJ02
PHASE voltage vas
to
I
- 5 .
-10
•15J
10mS
X10
- 2:
-iJ
■G3
10mS
REFERRED ROTOR F IE L D CURRENT TORQUE
X 10 '
10mS
X102
mS
Fig 4.16d V S I S y n c h r o n o u s Motor 10 Hz
187
phase current las PHASE VOLTAGE vosX10" X102
U)h<
mS20
- 10.
20
-2
LN.- i
RlS
REFERRED ROTOR F IE L D CURRENT TORQUE
X10
3
2
mS0 20
X I 0^ 5
i .
3 .
2.
10 15 20_» tr.S
Fig^.16e V S I Synchronous M o to r 50 Hz
188
PHASE CURRENT las PHASE VOLTAGE vasX102 X102
u>h<
mS0:1
- 1 0
-15J
to*-o>
mS
REFERRED ROTOR F IE LD CURRENT TORQUE
X10
3
2
X103
mS mS
F ig &.16f V S I Synchronous M otor 200 Hz
189
PHASE CURRENT los PHASE voltage v o s
X102 X10-15
10
toQ.C<mS
10
-5 ,
- 1 0
-1 5 J
(/)#-d■>
mS10
-2
- 1:
-6
L IN E v o l t a g e vobTORQUE
X1010,
- 5
-10
8X1I0'
10tnS
X I 0^5
1
3
2
1
mS010
X10
Fig A 17a CSI I n d u c t i o n Motor 10 Hz
190
PHASE CURRENT ios PHASE VOLTAGE vos
i/iI
X103 G,
1:
w 2:
- 2J
-iJ
•6i
—-Jla iè&
20mS
L IN E VOLTAGE vobTORQUE
X1010,
S.
- 5 :
- I 0 j
15 20roS
X I 03
20mS
Fig ^.17b CSI Induction Motor 50 Hz
191
PHARE CURRENT las PHASE voltage vosX)0" y ; 0-
coCLr
- 5 .
mS
-2
L IN E VOLTAGE vabX103
U)o
mS
-5
TORQUE
X1035,
3.
0,10 ]
mS
Fig A.17c CSI Induct ion Motor 200 Hz
192
PHASE CURRENT ias PHASE VOLTAGE vosX]0215
10.
toh<
X10
- 5
- 1 0
-15
mS
X10^
I—
5mS
10
- 2 .
- i
-6J
REFERRED ROTOR F IE LD CURRENT TORQUE
X I 0^
mS10
X10
X103
mS10
X10
Fig 4.17d CSI Synchronous M o t o r 10 Hz
193
X102PHASE CURRENT ios PHASE VOLTAGE vos
toè<
mS20
- 1 0
X1036,
u) 2
-2
-4
-6J
-5 A- 15 20
REFERRED ROTOR F IE LD CURRENT TORQUE
X103
mS20
X10
20mS
Fig ^.17e C S I S y n c h ro n o u s Motor 50 Hz
194
PHASF CURRENT la?X10- XI 0^
PHASE VOLTAGE vas
- 1 0
mS
to»-
- 2:
- 6 i
mS
REFERRED ROTOR F IE L D CURRENT TORQUE
X103 X10^
mS
1
3
2
1
mS0
Fig A.17f C S I Synchronous Motor 200 Hz
195
4 .7 Conclusions
The use of a vo ltage or cu rren t source in verter gives rise to conflicting
req u irem en ts fo r the m ach ine design . For a VSI fed m ach in e , a high
value of leakag e In ductance Is d es irab le to m in im ise the harm onic
c u rre n ts . W h e re a s , Ideally the leakag e Inductance of a CSI fed m otor
should be as low as possible to lim it the m agnitude of the com m utation
vo ltages . Th is In turn would red u ce the m ach in e Insulation stress and
en ab le in verte r com ponents of a low er rating to be used.
An In itia l step tow ards the reduction of leakag e reac tan ce In the Induction
m otor design c o n s id ered in this work, can be m ade by ensuring that the
slot depth is as sm all as possible. The e ffec t of a change In slot depth
Is read ily d em o n stra ted using the Induction m otor design program of
C h ap ter 2. T h e results of this excerc lse a re shown In Fig 4 . 18.
As the ro to r co n d u cto r cu rren t density for the Induction m otor design Is
below Its perm itted m axim um value . Fig 2. 13 . the rotor slot a rea may be
substan tia lly red u ced . If the slot depth Is reduced to the point w here the2
stator and ro tor co n d u cto r densities a re eq u a l. ( 8 .4 4 A /m m ) . the
rotor s u rface leakag e Inductance can be red u ced , theoretica lly by 50% .
Fig 4 . 18. In p ra c tic e this figure would be reduced to approxim ately 20%
due to the Inclusion of the slot opening In the leakage ca lcu lations. It can
also be seen that the d ecreas e In rotor leakag e Is accom pan ied by a fall
in the total m otor w eigh t, due to the fact that the core length can be
reduced s lightly . T h e re is also a slight in c rease In the overall power
facto r. Fig 4 . 18 also dem onstrates the e ffec t of varying the stator slot
depth . for equal stator and rotor conductor cu rren t densities. The core
length has been adjusted In each case to give the required power output
of 580 KW. and all o th er Input variab les to the design program rem ain
196
V s h a p e d r o t o r slot
t runcoted rotor slot
30
SR0
el l
0.8 800
□ moss
600
□ t. length
SSt . length
SR
2 200
0
60 5 0 60
stotor slot d e p th mm
70
Fig 4.18 E f f e c t of s t a t o r s lo t depth va r ia t ion
197
unchanged from those shown In Table 3. 2.
Variation o f the s ta to r slot depth produces changes In pole p itch, co re
length and ro tor slot depth . For an In c rease In slot depth above 50 m m .
the net e ffec t of th ese changes Is a reduction of the end winding leakage
factors kend and kend How ever this reduction Is offset by an LS LR
In crease in the s ta to r and rotor slot leakag e com ponent, which Is due
directly to the In c re a s e In slot depth . As the stator slot depth Is reduced .
the m axim um ava ilab le rotor slot a re a Is corresponding ly red u ced .
b ecause of the restric tions on the co re d iam ete r. This m eans that, the
depth to w hich the ro tor slot needs to be filled with conductor, d . hasR
to be In c re a s e d acco rd in g ly . Thus for a d e e p e r stator slot a shorter m otor
results with a co rresp o n d in g reduction In w eight, however as this Is also
acco m p an ied by a fa lling power fac to r, the cho ice of a 50 mm slot for the
traction designs co n s id ered Is fe lt to be Justified.
In this c h a p te r co m p u ter m odels have been presented that predict the
p erfo rm an ce of Induction and synchronous m ach ines whilst being fed from
a thyristor In verter of the constant voltage or constant cu rren t type. The
resulting system equations m ay be solved by readily availab le Iterative
techn iques .
The m odel proposed fo r the cu rren t source Inverter Includes the effect of
the DC link filte r by considering a regu lated vo ltage, ra ther than a
regu lated c u rre n t supply. From the output w aveform pred ictions, the size
of the In verter com ponents may be accu ra te ly dete rm in ed , and any area
of vo ltage stress Id en tified .
One a re a that req u ires to be looked at In m ore d e ta il. Is the m echanical
coupling to the locom otive ax le , because of the high harm onic torque
content at the m ach in e output shaft, particu larly at low speeds. If the
198
to rque h arm on ics co in c id e with the m ech an ica l resonances of the power
transm iss ion system , which inevitably som e w ill, a m ethod of curren t
m odulation m ay be requ ired to avoid p rob lem s. Som e methods availab le
to do this have been suggested in re fe re n c e s 42 and 43 .
199
4 . 8 A P PEN D IC ES
4. 8 . 1 List of p rincipal symbols used In the analysis of the voltage and cu rren t source Inverter
Induction machine
Rs
Is
Rr *
*R ’
Lm
stator phase resistance stator phase leakage inductance referred rotor phase resistance referred rotor phase leakage inductance
magnetising inductance
n
H
n
H
H
Synchronous machine
Rs
• s
Rp •
( p .
Lmd
^nq
stator phase resistance stator phase leakage inductance referred field winding resistance referred field winding leakage inductance direct aucis magnetising inductance (quadrature eucis magnetising inductance
Vas'vbs'Vcs machine phase voltagesias'ibs/ics machine phase currents
iaR'fibR'ficR' referred rotor currents(induction machine)
ip* referred field winding currents(synchronous machine)
n
H
n
H
H
H
V
AA
P
Up
e
Te
6«p
pole pairssteady rotor angular velocity angular position of rotor electro—magnetic torque torque angle
201
rad/secdegree
Nmdegree
y phase displacement between the fundamental
component of the machine phase current and the commutation point Iq degree
Vg DC source voltage V
DC link current A
RdC'Ldc DC link resistance and inductance D,HC inverter capacitance value F
Ceq equivalent capacitance value used ininverter model F
202
4 . 8 . 2 1 he induct ion machin e expressed in stator coordinates
T h e bas ic m a c h i n e m ode l is shown in Fig. A 4 . 1 . T h e rotor is r e p r e s e n t e d
by coi ls on the d and q axis, and is a s s u m e d to be t ravel l ing at a constant
a r b i t r a ry s p e e d wg.
Also
d©a dadt dt
1) V o l t a ge i n d u c e d in a stator coil
T h e vo l ta g e in d u c e d in a stator coi l m a y be e xp re s s e d as fol lows. For stator
coi l A,
Vas ^ Rs^aS + P * s i a S P^s'f’aS ( A 4 . 1 )
Reso lv ing the flux 0 3 5 into its d and q axis c o m p o n e n t s gives
4)aS ^ cos a - 0q sin a ( A 4 . 2 )
Subst i tut ing e q u a t i o n ( A4 . 2) into e q u a t i o n ( A 4 . 1 ) and di f ferent iat ing gives
Vas ^ RsiaS + P*siaS + P^S (‘f’d cos a - *q sin a)- WaNg ( 0 c| s in a + 0q cos a ) ( A 4 . 3 )
The d axis flux is g iven by.
14)d = - [Ns ias cos ©a + i-bS cos( ©a + 2 e ) + Ng i^s cos( ©a + e )
+ Nr i-aR cos /3 + Np ij p ( P + 2 e ) + Np i^p cos( p + e)]( A 4 . 4 a )
and the q axis by.
203
1*q = - L-Ns ias sin 0^ - Ng ibs sinfe* + 26) - Ng ics sin(6a + 6 )
- Np iaR sin P - Np i^p sin(/3 + 2e) - Np i^p sin(/3 + e)](A4.4b)
where Ng and Np are the effective number of stator and rotor turns2tt
respectively and e = — '
By the a p p l i c a t i o n ot the vol tage and c u r r e n t t r a n s fo r m s of F i g . A4. 2 , to the
rotor c u r r e n t s of e q u a t i o n s ( A4 . 4) the e xp re ss io n for the d and q axis flux
b e c o m e ,
4>d = p LNs ias cos ©a + Ng i^g cos( ©a + 2e) + Ng i^g cos( ©a e)
+ "R i f ;2 idR] (A4.5a)
and
14 > q = — [-Ng ias sin ©a ~ Ng i^g sin(©a *■ 2e ) - Ng i^g sin( ©a + e)
qR] (A4.5b)
It fol lows that
Ng(*d COS a - 0 q Sin a) = Lg ias cos( a - ©a)+ Dg ibg cos(a - ©a - 2e)
-f- Lg ics cos( a - ©a - ^) + M i^p cos a - } [ - M iqp sin a
(A4.6a)
and
Ng(*d s i n a + <t)q c o s a ) = Lg i a s s i n ( a - © a )
+ bg i j j g s i n ( a - ©a 2e )
L g i c s s i n ( a - ©a - f ) -t- - M i q p s i n a - ^ - M i q p c o s a
( A 4 . 6 b )204
Where Lg and M are the self and mutual inductance terms
n | N g N p
Lg = ~ and M -R R
If p h a s e a ' is a l ig n e d with the r e f e r e n c e axis then
for p h a s e a a = ©g
for p h a s e b a = ©g + 2 e
for p h a s e c a = © 3 + 6
On substi tut ion of e q u a t i o n ( A4 . 6 ) into e q u a t i o n ( A4 . 3 ) expressions for the
stator v o l ta g es m a y be found. T h e resu l t ing exp re ss io ns a re the first th ree
rows of the set of e q u a t ion s shown in Fig. A4. 3.
2) Vo l t age in d u c e d in a rotor coi l
T h e vol ta ge i n d u c e d in a rotor coi l m ay be e xp re s s e d as fol lows, for phas e
a'
VaR " Rr i a R + P^R + PNR4>aR ( A 4 . 7 )
w h ere 4>aR = cos /3 - <t>q s in p ( A 4 . 8 )
C o m b in in g e qu a t i o n s ( A 4 . 7) a nd ( A4 . 8 . ) e xp an d in g V3 P and 1 3 ^ using the
t ran s fo rm a t io n of F ig . ( A 4 . 2 ) . and d i f fe re nt ia t i ng gives:
1 1v q p COS p - V q p s i n p + — V q r = R p l i d R c o s p - i q p s i n p + — i o p ]
V2 /? .
L■*- P * R [ i d R cos p - I q p s i n p + — i o R j “ P * R [ i d R s i n p + i q p cos p \
V2
V T NplOd cos /3 + 4>q s i n P ] - p } [ -+ P y ~ NpLOd cos P + <t>q s i n P ] - /3 ^ - NplOq s i n /3 — <t>q cos /3‘J
Col lec t in g te rm s y ields
205
VjR = (%R + P*H)idR - #*%iqRiqR + P V :3 n2 Nr (pq ~ P M 2 <Jq
^qR - ( Rr + P*R)iqR + P*RÎdR + P 2 ^ V 7 Nr « d
VoR = (%R + P*R)ioR
(A4.9)
Substitution of equation (A 4 . 5) into equation (A 4 .9 ) gives expressions for
the rem ain ing axis vo ltages, vqR, Vqp and Vq R, and com pletes the set of
m achine equations shown in Fig. A4. 3.
3) Transform ation into stator coord inates
The set of equations shown in F ig . A 4 .3 are valid for the following
conditions, that
a) S tator phase a ' coil is a ligned with the fixed re ference axis
b) The rotor is travelling at a speed cur
c) The transform ed rotor is travelling at an arb itrary speed wg or at a
speed p with respect to the rotor
In o rder to bring the rotor to rest in the stator re fe ren ce fram e ©g is set to
zero and hence cug = 0 .
As ©a = 0R + /3. /3 = - 0R and p = - c u r .
The rotor which is now expressed in dqo coord inates must be transform ed
back to th ree phase coord inates.
205
T h e m a c h i n e e q u a t i o n s of Fig. A4. 3 m a y be exp re ssed in the fol lowing
g e n e r a l matr ix form
vg = Zss ig + ZgR idqo
“ ^RS Zdqo ^dqo(A4.10)
Where 2 3 5 . z g p . 2 3 3 , z^ q o are s u b -m atr ices of the total impedance matrix
2.
Using the t r a n s fo r m a t io n s of Fig. A4. 2 with /3 = 0 g ives.
VR ■ -V:
1 017 i
1 / 3 1
2 2 V2
1 / 3 1_ — _ — ---------2 2 V2
C Vdqo
VdR
^qR
VoR
id R 11
2
1
2
73 73iq R = > I 0 2 2
1 1 1ioR V2 72 72
Idqo = If Y C iR
Using the above re la t ionships equat ion ( A 4 10) b e c o m e s
207
Vg ^SS is > ^SR V4 C 1 r
V RT T
( A 4 . 1 1 )
(A4.12)3 Z r 5 i g + 3 C Zdqo C i p .
F r o m e q u a t i o n s ( A 4 11) and ( A 4 . 12 ) and using the fol lowing self and
m utua l i n d u c t a n c e re la t io n s h i p s , the c o m p l e t e set of stator re fe rr ed m a c h i n e
e q u a t ion s m a y be c o n s t r u c t e d .
M - l»s - L r -
w h e r e is the per p h a s e m a c h i n e m a g n e t i s i n g in d u c tan c e .
T h e c o m p l e t e d set of t r a n s fo r m e d p e r f o r m a n c e equat ions for the induct ion
m a c h i n e is shown in F i g . 4 . 1 . C h a p t e r 4.
4 ) E l e c t r o m a g n e t i c T o r q u e
T h e e l e c t r o m a g n e t i c t o r q u e d ev e l o p e d by an Induct ion motor having p pole
pairs is g iven by.
P T dL Te = I i ^ i
w h e r e L is the m a c h i n e in d u c t a n c e matr ix . F ig . A4. 4. On di f ferentat ion of
the L matr ix with r e s p e c t to e the exp ress io n for to rque b e c o m e s
P 1 T T'2 1 "S IR
P T ,Te - - Is L-Ir
0 L is
I."" 0 ÎR
T T T LIrIR -■ S - P Is
208
( A 4 . 1 3 )
where
sin Op sin(0p + 6 ) sin(9p + 2€)sin( 6p - 6 ) sin Or sin( Gp + 6)sin( 0p — 2<= ) sin(&R - 6) sin Gp
^aR
ibR
j-cR
Using the transfo rm ations of Fig, A4. 2 to transform to an arbitrary dq axis,
and bringing the fram e to rest with respect to the rotor i .e . p = - 6r then .
L ÎR =
0 1 0/3 12 2 0
/3 12 2 0
Again using the transfo rm ation of F ig . A4. 2 with p = 0 \o transform back into
th ree phase quan tities , and finally re ferring the Inductance coeffic ients to the
sta to r, the expression for the instantaneous torque is given as.
LmTe - - p ias(^bR' “ cR') + ibs(-iaR' + icR')
+ ics(^aR’ - ibR')
209
coil B
rotor phase o coil axis
s tato r p h a s e a coil ax is
( f ix e d )
Ref. a x is
coil A
Rs
'cS'cS
co i l C
'oS
Fig A4.1 Induct ion Machine Model
210
' 'dR ' 'd R
'qR
''oR • 'oR
-J \
cos p cos ( p * 2c ) c o s ( p ♦ c)
- s in p - sin 1 p * 2 (1 - sin ( p ♦ 6 )
1
/ 2
1 1
''oR • 'o R
''bR ' bR
'cR • 'cR
''oR ■ 'oR
' 'bR ■ 'bR
''cR • 'cR
-A
cos p - sin p 1
/2
cos ip ♦ 2 c ) - sin ( p ♦ 2c) 1
/ 2
cos 1 p ♦ €) - s in( p ♦ e) 1
''dR • 'dR
VqR . iqR
''oR 'oR
Fig A 4 .2 T r a n s f o r m a t i o n s f o r vol tage and c u r r e n t
211
.nl<
Z
n|r^
z
z »
ro| (M
q:fnjf'*
crrolfx
Z,mi<N
£T|
Z
ro|<N(K
nirsi
a• i .
z1<M
JDO
<C7»
212
I«
tri CM
213
C H A PTE R 5
EXPERIM ENTAL VERIFICATION OF THE CURRENT SOURCE INVERTER
MODEL
In this c h a p te r experim enta l results are shown to verify the com puter
m odel of the c u rren t source inverter described in chap ter 4.
5. 1 The test m ach in e and torque m easuring system
Fig 5. 1 shows a g e n e ra l view of the induction m ach ine used, and the DC
load m a c h in e . The stator fram e of the induction m achine was m ounted on
a fo rce m easu rin g tab le . The rotor was held between two bearing posts
at each end of the shaft. Fig 5. 2 . which a re rigidly bolted to a supporting
fram e. A lso m ounted on this fram e was the DC load m ach ine whose
speed was con tro lled by a W ard Leonard system . This arran g em en t
enab les the fo rces g en era ted between the stator and rotor m em bers to be
transm itted d irec tly to the force table for m easurem ent.
The induction m ach in e was of a four pole design , with a squirrel cage
rotor w ind in g , and was rated at 200V . 25A . A full description of this
m ach ine includ ing all the re levant d im ensions is given in Appendix 5 . 4 . 1 .
To d em o n stra te the validity of the induction m achine param eter
ca lcu la tio n s of C h ap ter 2 . the equivalent c ircu it values of this m achine
have been ca lcu la ted using the form ulae given in Appendix 2 . 7 . 2 . The
ca lcu la ted p aram ete rs a re shown together with the m easured values for
com parison in T ab le 5 .1 . The m easured values w ere obtained using the
norm al induction m ach in e open c ircu it and locked rotor tests. It can be
seen that th ere is good ag reem en t betw een the two sets of param eters .
Throughout this ch ap te r the equivalent c ircu it values obtained by
m easu rem en t a re used in the com puter m odel.
215
EooQ.
(UO
•OCO0;c!c8E
,it)313C
CNuiO)ll
3O>N0(LÇ'x:
1
(kcmCDLDgiLl
stator res is tan ce
re fe rred rotor res is tance stator leakag e inductance
re fe rred rotor leakage
inductancem agnetis ing Inductance
CALCULATED
0 .5 3 7 n
0 .7 2 3 5 n3. 035 mH
3 .8 7 3 mH
0 .1 1 1 5 H
MEASURED
0. 53 n
0 .8 3 6 6 n
6 . 908 mH 7. 53 mH
0 .1 2 3 7 H
T a b le 5 . 1 Induction m ach ine equivalent c ircu it param eters
The m easuring platform consists of a base fram e bolted to the m achine
bed , and an alum inium top p la te , with four 3 -c o m p o n e n t force
tran sd u cers fitted between them under a high prestress . Each transducer
consists of th ree quartz d iscs, each sensitive to p ressure in one of three
p re fe rred axis, x .y or z. The rotor is a ligned with the x axis. Forces In
this d irection w ere not m easured . The e lec trica l ch arg es yielded by the
fo rce tran sd u cers are converted via ch arg e am plifiers to voltages suitable
for d irec t m easu rem en t. The quartz transducers a re ab le to be preloaded
and still rem ain sensitive to sm all tim e varying com ponents of fo rce ,
provided of course the preload is not excessive. Thus by sum m ing the
ap p ro p ria te fo rces from each tran sd u cer and scaling by the torque arm
length In each c a s e , the torque produced by the m ach ine can be
m easured dynam ically . W hen the fo rces In the y and z d irection are
tran sferred to the cen tre of the ro to r. Fig 5 . 3 . the resu ltant m achine
torque is.
T = ( Fz - Fz )d + Fy hm (3 + 4 ) ( 1+ 2 ) (1 + 2 + 3 + 4 )
217
direction of rotation
l ine of action Fz
l ine o f a c t i o n F y
Torque Reaction
Fig 5 3 Two a x is forces for torque measurement
218
w here h Is the d is tance from the c e n tre of the rotor shaft to the line of
action of the fo rce tran sd u cer In the z d irec tio n , and d Is the d is tance
from the ce n tre of the shaft to the line of action of the fo rce transducer
In the y d irec tio n . In p rac tice the sum m ing operation was perform ed by
two operationa l am p lifie rs , the to rq u e arm lengths being scaled by suitable
resistor values.
219
5 . 2 Discussion of results
The DC link Inductor was chosen to be approxim ately ten tim es the total
m ach in e leakag e In d u c tan ce , to allow a reasonab ly constant cu rren t within
the m otor. The Inductor used has the following param eters ;
Rdc = 0 . 2 6 ohm s
Ldc = 80 mH
The com m utation cap ac ito rs used w ere 100 uF. 400V devices. As the
upper frequency lim it Is d eterm ined by the value of com m utation
c a p a c ito r, this restricts the highest test frequency to a theoretica l
m axim um of approxim ately 40 Hz. An upper frequency limit of 15 Hz was
th e re fo re Im posed to stay com fortably within the com m utation capabilities
of the Inverter.
A com parison of the ca lcu la ted and m easured m ach ine ch aracteris tics , for
diffe ren t loading cond itions. Is shown In Figs 5 . 4 . 5 . 5 and 5 . 6 . for
operating freq u en c ies of 5 . 1 0 and 15 Hz.
Typ ica lly , a steady state solution was obtained after 5 minutes of CPU
tim e. This requ ired approxim ately 50 Iterations over the 60 d eg ree
com puting perio d , and gave a solution that had an average erro r of less
than two dec im a l p laces for the five Input variab les , (cap ac ito r vo ltage,
phase and rotor c u r re n ts ) .
T h ere Is good a g re e m e n t between the com puted and m easured phase
220
"IT
UDO0b
1 (D
SdUlD
s
V SOI
2 21
s «o X T-
WN
uou
t/)E
WN
222
o<uE
S S8 8
S410A
oo
A qoA
223
3g
S410A
ou
A SDA
224
r 8
o
b
sduiD
Cl
SOI
2 2 5
T T
%WN
W N
226
388
8
S410A
§ § § §g g
A QDa
2 2 7
8 8 = 8 moA
>
8.
ooo
: g g
i/)E
SDA
228
s d iu D
V SOI
229
WN
W N
230
SJIOA
g g gg g
A q°A
231
s8 8S 410A
g
A SDA
232
cu rren ts , voltages and line to line voltages, the ca lcu lated values tending
to be slightly h igh er, due to a p robab le In accuracy In the value of motor
leakage inductance used In the m odel.
A g reem ent between the m agnitude and overall shape of the torque
pulsations is good , although superim posed upon the waveform is a
con sid erab le am ount of noise. T h e noise was gen era ted by the DC
m ach in e , and b ecause the force tab ie and the ioad m ach ine were rigidly
mounted to the sam e supporting f ra m e , was easiiy transmitted to the force
transducers . F iitering of this v ibrationai noise was considered but because
of its wide frequency sp read , and the fact that the ampiitude varied with
motor sp eed , it proved difficuit to provide a fiiter that wouid cope with
these conditions. Aiso, as certa in frequencies present within the noise
tended to excite resonances within the induction motor - force table
com bination , which artificiaiiy increased the m agnitude of the harmonics
present within the torque w aveform , it was difficuit to remove these
frequenc ies and still preserve the basic waveform shape.
A com parison of the caicu lated and m easured harm onic torques present
in the m ach ine output, is shown in T ab ie 5 . 2 . in view of the iimitations
m entioned above there is a reasonab le a g re e m e n t between the calculated
and m easured results.
H arm onic torque Nm (rm s )
6th 12th 18th
frequency caic . m eas . ca lc . m eas. ca lc . m eas.
5 2 7 . 5 2 2 .1 15. 9 11.0 10. 6 8 . 8
10 5 .7 9 4 . 8 6 2. 39 2. 59 1 .3 9 -
15 0 .2 1 0 .3 3 1 0. 051 0 . 106 - -
Tab le 5 . 2 H arm onic torques in output
233
ideally the fo rce table and Induction m ach ine should be m echanically
isolated from the load m ach in e , the only coupling then being via the
flexible drive between the rotor shafts. This should provide a more
accep tab le torque signal.
23A
5 .3 A PPEN D IC ES
5.3. 1 Details of test machine
STATOR
N u m b er of poles
N um b er of slots
Conductors per slot
(c o p p e r 0 .0 9 2 " LEWMEX)
Pole pitch
Coll pitch
Air gap d iam eter
C ore length
Slot pitch
Air gap length
C onductor o verh an g ( lo h s )
4
48
20
0 .1 7 9 6 M
0. 1498 M (1 0 slots)
0 .2 2 9 M
0 .1 0 1 6 M
0 .0 1 4 9 8 M
0 .0 0 1 M
0 .0 3 5 M
Ooo^
3 5 wedge
m m
STATOR SLOT
236
ROTOR
N u m b er of slots
C ore length
37
0. 105 M
End ring dimensions (b rass )
Resistivity assum ed approx. 0. 7 5 e -0 7 OhmM
D er 0 .1 7 7 M2
A er 0 .3 9 1 E - 0 3 M
Rotor bars located by wedging In end ring
and securing with pegs
6.4
rotor conductor2
7 . 6 mm
ROTOR SLOT
237
5 . 3 . 2 The cu rren t source Inverter
The c ircuit d iagram of the curren t source Inverter used to obtain the
experim enta l results described In this ch ap ter Is shown In Fig 4 . 9 . The
Inverter was fed with a variab le DC voltage supplied from a 3 phase varlac
and bridge rectif ie r via a series link inductor. Th e thyristors used in the
inverter w ere In ternational Rectifier type 81RK80 devices, and are rated at
125A rm s, 800V . These a re rectif ier g rad e ' thyristors, that is they have
a relatively long turn off time com pared to o ther devices available . The
turn off time of these devices is typically 80 /xS. The diodes used, are
again I. R. type 2 5 G 8 0 and a re rated at 95A rms.
A schem atic d iagram of the thyristor firing system is shown in Fig A 5 .1 .
The inverter thyristors a re fired in the co rrec t sequence by a shift
register. The outputs of the shift register a re turned into a burst of pulses
by the firing pulse gen era to rs . Fig A 5 .2 , and then amplified to a suitable
level by the firing circuits . Fig A5. 3. The firing circuits are isolated from
the thyristors by transform er coupling.
To ensure re liab le operation , multipulse firing Is used In the pulse
generators . The outputs of the shift reg is ter a re used to gate on' an
astable c ircuit. Fig A 5 . 2. The main advantage of this type of c ircuit is
that the first firing pulse occurs Im m ediately on arrival of the gating
signal. Schmitt NANO gates are used to provide a fast rising edge on the
firing pulses. The firing puises are then ampiified by a Dariington pair, to
drive a pulse tran s fo rm er. Fig A5 .3 . To avoid dam aging the thyristors the
voitage . cu rren t and instantaneous power appiied to the gate must faii
within the iimits specified by the m anufacturer. The use of a 15 voit power
238
supply ensures the voltage constra int Is m et. w hereas the current and
h en ce the power Is llmllted by R2. The d iode D2 is a fast recovery diode
and is provided to protect the gate from reverse bias, whiie R4 dissapates
the firing pulses if the anode cathode junction is reversed biased. C 1 and
R5 constitute a snubber circuit to limit the voitage transients which o ccur
during switching.
Prior to starting the inverter it is n eccesary to set up the initial conditions
both within the shift reg ister and the inverter commutation paths. An
external set switch is used to set up the logic pattern at the outputs of the
shift reg ister. W hen d ep ressed , this enab les the firing circuits for
thyristors T1 and T2. Due to the time constant of the load there Is
insufficient t ime for the curren t in the inverter legs to reach the holding
level of the thyristor before the end of the gate pulse. To overcom e this
problem a divert resistor is p laced around thyristors T1 and T2 to establish
the motor curren t. O nce this has been achieved the resistors are ab le to
be switched out, leaving the thyristors to take over the current path. The
inverter is now ready to run. The shift reg ister clock signal has to run at
six tim es the desired inverter operating frequency , ie one puise for one
thyristor com m utation . No externai methods a re required to set up the
required c h arg e distribution on the com m utation capacitors, as this is
done autom aticaiiy when curren t is estabiished in the motor windings via
thyristors T1 and T2.
239
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C HA PTER
SUMM ARY
It has been shown that even with a m odest value of RMS curren t density, it is
possible to design an induction or synchronous m ach ine weii within the weight
specification of 1600 kg and of a size that wiil fit between the w heelset of a
high speed locomotive. Of the three m ach ine types studied on induction or
salient poie synchronous m ach ine operating under a 'r ising voitage ' control
sch em e , seems to be the most attractive, from a weight point of view, in this
particular high speed traction application. The rising voltage' control schem e
produces lighter designs but does incur the penalty of requiring a h igher supply
capacity. This point will have to be borne in mind when the overall econom ic
and space requirem ents are considered.
T he most suitable induction and synchronous m ach in es a re s im ila r both in
physicai size and perform ance, and requ ire supplies of a s im ilar capacity.
However, because of the large air gap . the salient pole synchronous m achine
is approximately 25% lighter than the induction m ach ine . T h e synchronous
m ach in e does have the d isadvantage of requiring slip rings and a separa te field
supply, but these difficulties should not d iscount Its use for traction purposes.
In view of the reduced track loading that would result from being lighter.
If a heat transfer study of these two m ach ines w ere to be u n d ertaken , and it
proves that the use of h igher current densities a re feas ib le , a further reduction
In m achine weight would be possible.
C om puter models have been presented for a voltage and curren t so urce Inverter
operating In the 120 degree conduction m ode. An exact m odel of the current
source inverter has been developed In which the effect of the DC link
Inductance Is taken into account by considering the Inverter to be fed from an
Ideal voltage source and setting up the In v e r te r -m a c h in e equations accordingly .
243
An Induction m achine model has also been presented in which the rotor
variables have been transformed to the stator. This type of m ach ine
representation has certain advantages when used to m odel in v e r te r -m a c h in e
systems. This is b ecause, as the actual three phase voltages and currents are
used, the charges of state within the inverter a re easiiy observed. A lso, as
the stator and rotor currents are at the sam e frequency , the t im e taken to
evaluate a steady state solution is reduced .
T he in verter-m achine models are used to predict the p erfo rm an ce of the most
suitable induction and salient pole traction designs. Voltage and curren t
waveforms are shown, which allow the size of the inverter com ponents to be
accurate ly determ ined, and any a reas of voltage stress to be identified.
Torque waveforms are aiso predicted. These predictions m ake it possible to
Investigate the effect of harm onic torques on the power transm ission system,
and identify any source of m echan ica l reson an ce that they may induce.
M easured results are presented for a current source inverter fed laboratory
Induction machine. The ag reem en t between the m easured and predicted
waveforms including torque pulsations Is thought to be accep tab le enough to
verify the com puter model.
244
C H A PTER 7
R EFER EN C ES
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Digiiai com puters as an aid in e iec tn ca l m ach ine design Trans. AIEE
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[5] R eece A B J , C ha lm ers 8 J
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Application of digital com puters to tran sfo rm er design Trans AIEE Aug
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[8] Sharpley W A , Oldfield J V
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'Optimised p aram ete r analysis of an induction m achine Trans AIEE Nov
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[17] Barwell F T
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[19] Roffler M
'Class Am 6/6 and Class Ee 6/6 d iesel and convertor locomotives of the
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[20] S c h a e r R. Schmid A and S e g e r T
Articulated Tram s '2000 ser ies ' with three phase drive of the Zurich
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[22] Charlton W
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[23] Novotny D W
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'Analysis and com parison of two types of square wave inverter drives '
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251
